Whether a functional which preserves maximum for comonotone functions is monotone?

Question

Let $X$ be a compactum (compact Hausdorff space). By $C(X,[0,1])$ we denote the space of continuous functions endowed with the sup-norm We also consider the natural lattice operations $\vee$ and $\wedge$ on $C(X,[0,1])$ and the corresponding order. We call two functions $\varphi$, $\psi\in C(X,[0,1])$ comonotone (or equiordered) if $(\varphi(x_1)-\varphi(x_2))\cdot(\psi(x_1)-\psi(x_2))\ge 0$ for each $x_1$, $x_2\in X$. Let us remark that a constant function is comonotone to any function $\psi\in C(X,[0,1])$.

We say that a functional $\mu:C(X,[0,1])\to[0,1]$ (by a functional I mean only a map without any additional property) preserves $\vee$ for comonotone functions if $\mu(\psi\vee\varphi)=\mu(\psi)\vee\mu(\varphi)$ for each comonotone functions $\varphi$, $\psi\in C(X,[0,1])$. $\mu$ is called monotone if $\mu(\varphi)\le\mu(\psi)$ for each functions $\varphi$, $\psi\in C(X,[0,1])$ such that $\varphi\le\psi$.

The question is whether a functional $\mu:C(X,[0,1])\to[0,1]$ which preserves $\vee$ for comonotone functions is monotone? I can prove it for finite $X$. But I have problems in general case.

Thank you in advance for any information.

Answer 1

I have a partial answer: it follows from results of arXiv:2208.07152 that a normed homogeneuos respect any t-norm functional which preserves max for comonotone functions is monotone. But the general question is still open.