Tropical disk counting reconstruction conjecture for wall structures

Let $\sigma\in\mathscr{P}_{\operatorname{max}}$, let $v\in\sigma$, and let $P_{v,\sigma}$ be the monoid determined by the multivalued piecewise linear function $\varphi$. For $k\in\mathbb{N}$, define the counting function by

$$\log f_{\sigma,x}:=\sum_{\substack{\overline m\in\Lambda_{x,B},\\ \ell\leq k}}\sum_{\mathbf w}\operatorname{length}(\overline m)\#\mathcal M_0(\mathbf w,\overline m,\ell)^{\operatorname{gen}}z^{\overline m}t^\ell\prod_\tau t_\tau^{|w_\tau|}$$

in $\Bbbk[P_{v,\sigma}]\otimes_\Bbbk R^k$. After reduction modulo $(t^{k+1})$, suppose this counting polynomial stabilizes in $\mathbf w$ and maps to $R_{g,\sigma}^k$ by $t_\tau\mapsto1$. Define

$$\mathfrak{p}^k[x]:=\overline{\left\{y\in\sigma\setminus\partial\sigma\mid \log f_{\sigma,y}=\log f_{\sigma,x}\neq0\in R_{g,\sigma}^k\right\}}.$$

Tropical disk counting reconstruction conjecture. For every $k\in\mathbb N$, the sets $\mathfrak{p}^k[x]$ are either empty or polyhedral subsets of codimension at least one, and, up to refinement and addition of trivial walls, the sets $\mathfrak{p}^k[x]$ with their functions $\log f_{\sigma,y}$ reproduce the consistent wall structure $\mathscr{S}_k$ constructed in the cited algorithm. This conjecture identifies the consistent wall structure with loci on which the tropical disk-counting functions are constant.