The Schober reconstruction conjecture from versal harmonic maps

Let $\Sigma$ be a spectral cover of a Riemann surface $X$, let $\phi$ be the multivalued differential corresponding to $\Sigma$, and let $\mathfrak{S}$ be the perverse Schober on $X$ associated with the conic bundle. For each $\theta\in\mathbb{R}$, let $h_\theta$ be a versal harmonic $e^{i\theta}\phi$-map.

Schober reconstruction conjecture. The Schober $\mathfrak{S}$, together with a stability structure on it, can be recovered from the family

$$\{h_\theta\}_{\theta\in\mathbb{R}}$$

of versal harmonic $e^{i\theta}\phi$-maps.

This proposes a geometric reconstruction of categorical stability data from the family of harmonic maps. The source explicitly says that a general theory for buildings of arbitrary rank remains an open problem, and gives no resolution of this reconstruction claim.