Markov type monotonicity under finite-fiber submetries

Let $X$ and $Y$ be metric spaces. A map $\chi:X\rightarrow Y$ is a submetry if, for every $x\in X$ and every $r>0$, it satisfies

$$\chi(B(x,r))=B(\chi(x),r),$$

where $B(x,r)$ is the closed ball centered at $x$ with radius $r$. Assume that every fiber $\chi^{-1}(y)$ is finite.

Finite-fiber submetry conjecture. Under these assumptions,

$$M_2(X)\ge M_2(Y).$$

This conjecture would provide a uniform approach to the preceding results on Markov type constants for coverings and quotients. The supplied text does not state whether the conjecture has been resolved.