What did Whitebeard and Katakuri aspire for? To protect their family.
Mihawk aspired to be the strongest swordsman as a youngster and then ended up fulfilling his dream.
He is literally a narrative foil for Zoro - "Marine Hunter"/"Pirate Hunter".
The moment Mihawk was shown to be the Dream of Zoro - he was 100% confirmed a Conqueror.
I think you don't understand what I'm saying.
I'm not saying that Mihawk won't get the conqueror haki. I'm saying that there is no narrative purpose right now to give him.
Oda can decide just like Katakuri and Rayleigh to give Mihawk the conqueror for no "conqueroresque" reasons. But right now, there is no reason to think that Mihawk will have it. Simply because Conqueror was establish to be granted by a set of parameters. The main one being that one need to have the will of an aspiring King/Queen. Meaning, a conqueror.
Note: There is a narrative reasons to give conqueror to both rayleigh and Katakuri.
- For Rayleigh it was to settle the real apparition of Haki and to make a narrative stance about the power of the second of the pirate King
- For Katakuri it was to put Katakuri on Luffy's level as an equal (everything in Katakuri mirrors Luffy narratively)
Once Oda put the conqueror haki on the page, we can use our imagination to find a reason for the presence of that haki. But right now.
Mihawk doesn't have conqueror and do not need to have it.