Sure that’s part of it but even more so than that.
1.chapter 52 title: The Oath Mihawk and zoro make promises to each other Mihawk will await Zoro at the top , and Zoro promises that from now till the day I BEAT HIM to become the wss I will never lose so Zoro only becomes the wss when he defeats Mihawk plus Mihawk at the end say he wants wants to encounter both Zoro and luffy again someday.
2. Doesn’t matter that they trained with each other, before accepting him as his pupil Mihawk says to Zoro :so you want me to train you in the sword so that one day you can take my head.
3. End of 1033 zoro reflects on his promise to his captain… it’s still ongoing the fight will happen imo at least. I know some on here say it won’t but I honestly just think most are trolling or something.