Tropicalization surjectivity conjecture for splitting type loci

Let $\Gamma$ be a $k$-gonal chain of loops, and let $C$ be a curve of genus $g$ and gonality $k$ over a nonarchimedean field $K$ with skeleton $\Gamma$. The tropicalization map is

$$\operatorname{Trop}:\overline{W}^{\bm{\mu}}(C)\longrightarrow\overline{W}^{\bm{\mu}}(\Gamma).$$

Tropicalization surjectivity conjecture. The map $\operatorname{Trop}$ is surjective. Moreover, if $g=\lvert\bm{\mu}\rvert$, then it is a bijection.

This conjecture would imply the numerical class conjecture above. It remains open in many cases where the numerical class conjecture is known to hold.