The generic polygon image conjecture for polynomial almost-complex curves

Let $\mathsf{TS}_{k}$ be the moduli space of the polynomial almost-complex curves under consideration, let $\mathsf{TP}_{k+6}$ be the moduli space of marked polygons with $k+6$ vertices modulo the $\mathsf{G}_2'$-action, and let $\mathsf{TP}^{\mathsf{gen}}_{k+6}$ denote its generic locus, consisting of marked polygons for which $d_3(p_i,p_l)=3$ whenever the induced graph distance between $p_i$ and $p_l$ is at least $3$. Consider the map $\alpha: \mathsf{TS}_{k}\rightarrow\mathsf{TP}_{k+6}$.

Generic polygon image conjecture. The map $\alpha: \mathsf{TS}_{k}\rightarrow\mathsf{TP}_{k+6}$ has image in $\mathsf{TP}^{\mathsf{gen}}_{k+6}$ and is a homeomorphism onto its image.

The conjecture is motivated by the matching dimension $2(k-1)$ of $\mathsf{TS}_{k}$ and the generic polygon locus, as well as by analogous homeomorphisms between moduli spaces of holomorphic differentials and polygon moduli spaces obtained from asymptotic boundaries. The supplied text does not state whether this conjecture has been resolved.