No. If gryphon was rogers not rays.
Then if they fought mihawk could have broke rays sword which is why he only has the other one now.
I think with enough Haki any sword can be a black sword and Prime Ray can be enough to take down Mihawk ... and we don't even know if Mihawk had his blade of 12 or he was using a normal sword
but beating Ray CAN BE a case ....
let's say Ray was giving up on life ... and he meet a Young Mihawk who want to fight him cause Ray is a great swordsman
Mihawk win ... simply cause Ray don't want to fight back ....
the world sees it as a Young man beat a legend
but Mihawk don't really enjoy his victory ...
time later he talk to Ray and Ray would tell him if you want a real match go and find Shanks cause I'm done fighting
therefore Shanks and Mihawk legendary fights begin
if I were Oda I would do this
p.s
Mihawk was famous BEFORE GREAT AGE ... so the time line is HARD to fix for that time
there is only 1 years time gap here before Ray be on himself and start of great era