Tropical wall-crossing conjecture for reduced open Gromov–Witten invariants

Let $\gamma$ be a primitive charge. Suppose that $u$ crosses a wall consisting of relative classes $\gamma_i$, for $i=1,\ldots,n$. For each $i$, let $\mathbf{w}_i=(w_{i1},\ldots,w_{il_i})\in\mathbb{Z}_{\geq 0}^{l_i}$ and set $|\mathbf{w}_i|=\sum_{k=1}^{l_i}w_{ik}$. Write $\mathbf{w}=(\mathbf{w}_1,\ldots,\mathbf{w}_n)$, let $\operatorname{Aut}(\mathbf{w})$ denote its automorphism group, and let $N^{\operatorname{trop}}(\mathbf{w})$ be the integer count of tropical discs associated with $\mathbf{w}$. Wall-crossing conjecture. The jump of the reduced open Gromov–Witten invariant satisfies

$$\Delta\tilde{\Omega}(d\gamma)=\sum_{\mathbf{w}:\,\sum|\mathbf{w}_i|\gamma_i=d\gamma}\frac{N^{\operatorname{trop}}(\mathbf{w})}{|\operatorname{Aut}(\mathbf{w})|}\left(\prod_{1\leq i\leq n,\,1\leq j\leq l_i}\tilde{\Omega}(w_{ij}\gamma_i)\right).$$

The formula is expected to be equivalent to the Kontsevich–Soibelman wall-crossing formula and expresses the jump through tropical-disc counts. The source does not provide a proof, and the tropical multiplicities are attributed to Gross–Pandharipande–Siebert and the cited review.