Consider the following min-max problem

$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$

where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\inf_{x\in M} \sup_{y\in N} F(x,y)=\sup_{y\in N} \inf_{x\in M}F(x,y)$ if $M\subset \mathbb R^m$ and $N\subset\mathbb R^n$ are both compact?

PS: To the best of my knowledge, the reference on Min-Max theorem is from M. Sion : https://msp.org/pjm/1958/8-1/pjm-v8-n1-p14-p.pdf However, the convexity of $x\mapsto F(x,y)$ is missing in my case. Any comments or references are highly appreciated!

PS2: Thank Nik Weaver for the counterexample and