Thing is Shanks has conquerors confirmed, while Mihawk doesn't.
The argument that his MF showings weren't good enough is feeble since he was clearly most likely struck with PIS. His slash directed towards Shirohige palpably casual despite that should not of been the case.
And other characters ending up under the same doom... Kizaru failing to kill Luffy despite his goal, Mihawk too...
Mihawk also stating that he reputes Shanks not to constitute a real match to him, I don't get why it would be likely that Mihawk was mistaken that wise.
A non CoC user can beat a CoC user if the complexive power is good enough but since Shanks' CoC looks so powerful, at least potentially, I expect Mihawk to likely possess it too, unless his overall CoA and CoO and skill outputs are overall that good on top of speed and strenght complexively, that he can do without.