no?
mohawk is the one who talked about settling. meaning he's the one who needs to beat the rat.
That doesn't make any sense. If that's the case why doesn't Mihawk challenge Shanks to prove he is stronger. That would also mean that Shanks is the WSS. All of that goes against the manga and the information we have.
The most logical explanation is Mihawk promise Shanks he would seek him out and challenge him after Shanks became a yonko to see if Shanks could tie the score. After fighting one arm yonko Shanks, Mihawk realized Shanks could no longer keep up with him and was no longer a worthy opponent so they stopped the fight. Remember, Rocks had trouble fighting when he was limited just one arm.
Now, Shanks is hoping Mihawk will one day give him the opportunity to prove he's worthy and tie the score.