Bounded convex decomposition conjecture for rotation sets of surface homeomorphisms

Let $S$ be a surface of genus $g\ge 2$, and let $f\in\operatorname{Homeo}_0(S)$. Write $\operatorname{rot}(f)$ for the rotation set of $f$.

Bounded convex decomposition conjecture. There exists a map

$$m:\mathbf{N}_{\ge 2}\to\mathbf{N}$$

such that, for every surface $S$ of genus $g\ge 2$ and every $f\in\operatorname{Homeo}_0(S)$, the rotation set $\operatorname{rot}(f)$ is the union of at most $m(g)$ convex sets. Moreover,

$$m(g)\le 2^{5g-5}.$$

This conjecture suggests that rotation sets of arbitrary surface homeomorphisms admit a uniformly bounded decomposition into convex pieces depending only on the genus. The supplied text gives no resolution status beyond posing the statement as a suggested persistence of the preceding theorem.