The broken-line conjecture for holomorphic cylinder counts

Let $B$ be the base of the log Calabi–Yau surface, and let $\beta\in\operatorname{NE}(Y)$. Let $\gamma$ be a broken line in $B$ for the canonical scattering diagram, and choose a negative number $\alpha$ such that $(-\infty,\alpha]$ is a proper subset of the unique unbounded domain of linearity of $\gamma$. The restriction of $\gamma$ to $[\alpha,0]$ gives a spine $L_\gamma$ in $B$. If the monomial attached to the last domain of linearity of $\gamma$ is $c\cdot z^q$, with $q=\beta+\lambda$ for $\lambda\in\Lambda_{\gamma(0)}$, then the broken-line conjecture. The number $N(L_\gamma,\beta)$ of holomorphic cylinders associated to the spine $L_\gamma$ and the curve class $\beta$ is equal to $c$. The conjecture gives a precise relation between holomorphic cylinder counts and broken lines, suggesting that the positivity, integrality and gluing results for cylinder counts reflect the combinatorics of the canonical scattering diagram. The source provides no resolution of this conjecture.