Here's my two cents - not on min wage specifically but on my read of the bigger landscape.
I think the fundamental competition in a free market is between labor and capital. The rising tide of economic growth should be lifting both where there is no clear winner or loser. Government should be a neutral party erring on the side of whoever has the least power. But humans don't work like that.
The problem is that this competition happens at an abstracted level with government regulators, politicians, special interest groups, etc. It's not self-correcting except over very long time spans.
We saw massive power for labor as we began regulating industries - child labor laws, 40 hour work weeks, minimum wage laws, and then post-depression era social programs. The power of labor peaked and the power of capital grew. Now we've had a 40+ year cycle where politicians responded to financial interests rather than the interests of the working class.
Then, 2016 hits and we see populism left and right. What both MAGA and Bernie/AOC world have in common is a belief that Washington DC works for the benefit of the wealthy and well connected, not them.
It's my opinion that we are on the verge of witnessing a multi-decadal shift. If the populists stay divided on economic issues because of cultural issues, then labor will continue to lose. But if they unite on economic issues, the tide shifts and we enter a multi-decade cycle where begin moving the other way.
Realize this - the core of Trump's base supports UBI like payments, increased minimum wage, etc - same as the left. There's a reason populist-wannabe Josh Hawley stood with Bernie in support of $2k payments. Economic populism is creeping into the right as well. We may not be that far from both parties supporting economic populism while fighting over cultural issues.
$15 dollars was a 60% winner here in Florida in a Trump state. This is bi-partisan winning position and I think the shift is underway.