The spectral-network stability conjecture for Fukaya categories with coefficients

Let $S$) be an oriented surface with area form and marked points $M\subset S$, and let $\mathcal E$ be a schober of triangulated categories over $\mathbf k$ on $S$ without singular fibers. Assume that the Fukaya category with coefficients $\mathcal F(S,M;\mathcal E)$ is defined, and write $\mathbf K$ for the Novikov field. Let $\sigma$ be a holomorphic family of stability conditions on $\mathcal E$, and let $Z$ be the central charge given by the graph formula.

Spectral-network stability conjecture. There is a stability condition on the homotopy category of $\mathcal F(S,M;\mathcal E)_{\mathbf K}$ with central charge $Z$, whose semistable objects of phase $\phi$ are precisely those objects having a spectral network representative of phase $\phi$.

This conjecture is part of a program suggested by Kontsevich and is verified in several examples, including disks with three and six marked points on the boundary. Precise convexity or completeness conditions at infinity are needed in the non-compact case.