If Shiliew killed Mihawk how exactly that would make Shiliew the WSS?
Let me explain, from what I gather in the One Piece community, the WSS’s title only apply to:
-People without devil fruit (which is why Fujitora is not concern by it)
-People who are actively seeking to be WSS ( which is why Shanks is not concern by it)
And we know that :
-Shiliew has a devil fruit
-Shiliew never shown any interest of being the WSS
Furthermore, the WSS’s title is skill based title so if Shiliew beats Mihawk thanks to dirty tricks (like some people thinks) he would not prove himself to be the skillest of the two
Therefore, based on the aforementioned elements, even if Shiliew beats/kills Mihawk he will not be consider to be the new WSS
TL\DR: Shiliew will not be the WSS even if he beats/kills Mihawk so he will not beats/kills Mihawk.