Stability-space uniformization conjecture for generalized root systems of affine ADE type

Let $A=(a_1,a_2,a_3)$ satisfy $\chi_A>0$. Let ${\mathcal X}_A$ be the associated parameter space, $\widehat{W}_A$ its extended Weyl group, ${\mathcal D}_A$ the associated triangulated category, and ${\rm Stab}({\mathcal D}_A)$ its stability-condition space. Let ${\mathcal X}_A^{\mathrm{reg}}$ be the regular locus, let ${\mathcal D}^{\mathbb X}_A$ be the associated $\mathbb X$-Calabi–Yau category, and let $\mathrm{QStab}_s({\mathcal D}^{\mathbb X}_A)$ denote the space of $q$-stability conditions with respect to $s\in\mathbb C$, with principal component $\mathrm{QStab}^{\circ}_s({\mathcal D}^{\mathbb X}_A)$. Stability-space uniformization conjecture. There are biholomorphisms

$${\mathcal X}_A \big/ \widehat{W}_A \cong {\rm Stab}({\mathcal D}_A) \big/ {\rm ST}_1({\mathcal D}_A)$$

and, for every $s\in\mathbb C$ with $\operatorname{Re}(s)\geq2$,

$${\mathcal X}_A^{\mathrm{reg}} \big/ \widehat{W}_A \cong \mathrm{QStab}^{\circ}_s({\mathcal D}^{\mathbb X}_A) \big/ \widehat{{\rm ST}}({\mathcal D}^{\mathbb X}_A).$$

These predicted biholomorphisms relate the deformation space and its regular quotient to ordinary and $q$-stability spaces, respectively, as expected from mirror symmetry. The supplied text gives no resolution of either assertion.