IDK
Mihawk isn't relevant to the plot so obviously Oda won't bother to tell us about his fight; we might learn more later on when Mihawk became relevant
You understand this argument made no sense right? It's like we never been told on challenges on BM before WCI because this wasn't relevant
Hell, being WSS isn't even the only reason ppl might try challenge Mihawk, remember how Luffy and Zoro wanted to fight Sir Crocodile once they heard he is Warlord? Or how some Supernove (Luffy, Kid, Law, Apoo and Hawkins) want to challenge Kizaru for being an Admiral (well Hawkins did this because the cards...).
Being in important position is a reason for novice to want to fight you, so they can prove themself.
I don't ses why Mihawk be any diffrent😑