Lattice-preserving invariance conjecture for higher-genus Göttsche-Schroeter invariants

Let $\Delta$ and $\Delta'$ be two $h$-transverse polygons. A lattice-preserving transformation is an element of the affine group of $\mathbb{R}^2$ preserving the lattice $\mathbb{Z}^2$. Let $g_{\max}(\Delta)$ and $s_{\max}(\Delta,g)$ denote the maximal genus and maximal descendant parameter for which the combinatorial invariant $G_g(\Delta,s)$ is defined. Lattice-preserving invariance conjecture. If there exists a lattice-preserving transformation $f$ such that $f(\Delta)=\Delta'$, then for any $g\in\{0,\dots,g_{\max}(\Delta)\}$ and $s\in\{0,\dots,s_{\max}(\Delta,g)\}$ one has

$$G_g(\Delta,s)=G_g(\Delta',s).$$

The conjecture is supported by the computed examples and is asymptotically true in genus $1$ for nonsingular horizontal polygons by results cited in the paper, but it is not established in general.