Spectral-network pre-slicing and stability-condition conjecture

Let $\mathcal{F}_{f}(z)$ be the Fukaya category associated with the point $z$, and let $\mathcal{P}_{ij}(\theta)$ be the subcategory consisting of Lagrangian branes obtained from spectral networks of phase $\theta$ whose edge incident to $z$ is labeled by $ij$, for distinct objects $i,j\in Ob(\mathbb{V}_{\mathcal{S}})$. Set

$$\mathcal{P}=\{\mathcal{P}_{ij}(\theta)\}.$$

Spectral-network stability-condition conjecture. The collection $\mathcal{P}$ forms a pre-slicing of the standard $\mathbb{V}_{\mathcal{S}}$-collection, and $(Z,\mathcal{P})$ is a $\mathbb{V}_{\mathcal{S}}$-stability condition. This asserts that the spectral-network construction supplies the slicing data needed for a stability condition in the associated two-dimensional theory. The supplied text does not state whether the claim has been proved or remains open.