The claim I'm about to make is as researched as anything can be.
So please, read carefully and take in all information before rushing to reply, that is all I ask for.
Without further reading, let's deal with all arguments:
1. ARMAMENT HAKI PLAYS A MORE IMPORTANT ROLE IN SWORDSMANSHIP
This is simple, throughout the series, whenever swordmanship is mentioned, it always will come accompanied with new information on Armament Haki, and Armament Haki alone. Conqueror's has never played a role, and this can be seen throughout the one thousand and more chapters written by Oda's hand:
Ryou Explanation:
When Ryou was first explained, it was purely presented as a different name for Armament Haki, or Advanced Armament as some people call it.
This is coming from the land of Samurai, the leaders in swordsmanship, specifically naming and refering to Armament Haki as opposed to Conqueror's.
When Luffy succesfully mastered Advanced Armament Haki, Hyougoro concluded the lesson on Ryou as finalised.
Black Blades:
Mihawk's first and most important lesson, was on the nature of Armament Haki and Swordsmanship. He clearly outlines and points out that to create a black blade, armament Haki is needed. His is the strongest sword in the world, so we can take his advice as factual. Furthermore, Zoro's training specifically focused and dealt with Armament Haki, and as swordsmanship, a huge strain is placed upon it.
Such as not breaking blades.
Slicing things which are normally not sliceable.
Or creating Black Blades.
All things, tied with Armament Haki, which have shown in the story to have a deep tie to swordsmanship, and NOT Conqueror's.
Koushiro's Lesson:
One of the most talked about lessons in swordsmanship comes from Koushiro, in which he explains to Zoro how a swordsman must develop the ability to cut anything he wants, he thinks about this lesson in the context of his fight against Mr. 1, and develops, as the FIRST crewmember too, Armament Haki for the first time, enabling to cut through Mr 1's steel.
This also goes back to Hyougoro's lesson on Ryou, with similar phrasing, once again emphasing Armament Haki.
Unless you want to make the argument that what Zoro actually developed here was Conqueror's, it is undoubtedly an Armament Haki focus in the story, tied deeply with swordsmanship.
I could go deeper into it, but it would just be hammering the same point over again.
Swordsmanship is tied deeply with Armament Haki, and everytime an important piece of lore on it is mentioned whether it be Mihawk's lessons or Black Blades, there is always a tie to Armament Haki. Conqueror's Haki has yet to be mentioned at all as having any importance to swordsmanship.
But this of course raises the very important question:
2. WHAT ABOUT KING OF HELL??
King of Hell is without discussion, the application of Conqueror's Haki to swordsmanship.
Zoro was acknowledged by King as having Conqueror's, and thus procceeded to use it to defeat him, what does this mean for the future of swordsmanship?
To explain further, we have to do something a lot of people don't normally do,
we must read the whole context.
King isn't a Swordsman:
At the very start of the fight, we got this short scene, in which Zoro is suprised by King's moveset.
He not only relies on his sword to fight, but also punches and kicks too.
And after Zoro's shocks, by his own words, he exclaims that that must mean King is NOT a swordsman.
This is something that shouldn't be up for discussion, Zoro himself is making the statement that, since King plays under a different set of rules as normal swordsmanship, he is therefore not considered one.
But it doesn't end there, because immediately after he makes the exclamation that he too might rely on techniques outside of swordsmanship to win this fight.
He literally makes the foreshadowing, that he will win not through swordsmanship, but BY ANY MEANS POSSIBLE.
Kozaburo's Lesson:
During his fight with King, Zoro has a brief flashback in which he recalls a lesson by Enma's swordsmith:
"Swords are created to kill people".
Now, for those that are interested, you can look into the story of "Masamune and Muramasa", the two greatest swordsmiths in Japan.
Because this point is a carbon copy of that story, and it shows very well.
If Koushiro's lesson is "A sword that destroys is no sword at all", then it stands out that it's a COMPLETE OPPOSITE to Kozaburo's lesson, who believes swords are made only to kill and take, while Koushiro believes swords exist to protect and not-cut.
What does this mean for the fight? It means that during this fight, Zoro is recalling a lesson that is taking him away from true swordsmanship, as is shown in the Masamune and Muramasa story, he is leaning into the nature of swords as "Killing Machines", which is what he described King as being before.
During this fight, he is as stated by him, foregoing the path of true swordsmanship.
King of Hell, and Zoro's Dream:
What does this mean for King of Hell?
It means that KoH is not "swordsmanship".
And it is explained by Zoro's ideal that was explained all the way back in Thriller Bark, he is willing to sacrifice himself, and thus his dream, for the sake of Luffy's dream.
Zoro favours Luffy infinitely above himself, and thus would do anything to help him achieve this.
To conclude, Zoro starts by pointing that King is not a swordsman, and if he is to win the fight, he will have to abandon swordsmanship too and become a "Killer Machine".
What happens? He has a flashback in which Kouzaburo teaches him a lesson which contradicts THE most important lesson in swordsmanship yet, he exclaims that swords only exist to kill and destroy.
And thus, he enables King of Hell, this is him "biting King's throat off", as once prophesized, and abandoning Armament Haki the technique deeply tied to swordsmanship, in exchange for Conqueror's Haki, something else entirely.
3. WHY CAN'T CONQUEROR'S BE PART OF SWORDSMANSHIP
And all of this would make sense if it weren't for the simple questions:
"Why can't Conqueror's be used with Swordsmanship?"
"What does Conqueror's mean then?"
Or even
"But Mihawk MUST have Conqueror's!"
So here I will try to explain my reasoning as best as possible.
Mihawk's Statement:
Mihawk makes a very clear rationalization:
That Luffy's power to have people crowd to him is the most formidable of all.
This is without a doubt talking about the innateness of Conqueror's Haki, the Quality of a King, the power to amass followers.
Mihawk in this exchange, is making the delineation between what it means to be a Conqueror "Having people follow you", and what it means to be a swordsman "Techniques".
He, as World's Strongest Swordsman, understands this, because he still holds the dream of becoming Pirate King too, but was unable to, as he lacks the necessary qualities to be followed and lead.
Lone Wolf vs Kings:
It is no secret that Mihawk is an extremely lonely individual.
He lives alone in an island, mantains next to no contact with other people, and despite being WSS's has gained zero followers or pupils, except from the one that wanted to take him down eventually as sworn enemy.
Other characters like Luffy, Roger or Whitebeard, have shown a complete polar opposite personality, characters which have displayed extreme proficiency in Conqueror's Haki.
They have a following, they are beloved, people mourn and cry for them, and they are trusty leaders to the countless masses they make wherever they go. As the manga points out, these are the proper qualities of a Conqueror, and someone to be King.
So to finish this last point, Mihawk most probably has Conqueror's Haki, but it is in no way a vital part of becoming a swordsman, thus it is meaningless to argue.
He correctly pointed out Luffy's ability to make friends, an ability he lacks, and is deeply tied to being a King.
Joyboy wanted to be "King".
Luffy wants to be "King of the Pirates".
Zoro became "King of Hell", when he abandoned his swordsmanship.
And Mihawk? He is the "World's Greatest Swordsman".
As Mihawk has pointed out, the path to becoming a King is far far different and more arduous than the path to become the World's Greatest.
A path he once tried to take, and abandoned.
