Wall-correspondence conjecture for orbifold Hilbert-scheme moduli

Let $s\in HH_0([E/\Gamma])$, let $S$ be the deformation base, and let $M_{\underline\sigma}(v;\mathcal O_{D_\infty})$ be the relative moduli spaces associated to geometric relative stability conditions. Let $N^1(X_R^{[n]}/B)_{\mathbb R,>0}$ denote the relative divisor classes pairing positively with a curve in a generic fiber. Wall-correspondence conjecture. There is a nonempty subspace $U\subset\operatorname{Stab}(\mathbb P_{[E/\Gamma]}(\Xi)/S)$ of geometric relative stability conditions and a surjective map

$$\ell:U\longrightarrow N^1(X_R^{[n]}/B)_{\mathbb R,>0}$$

such that the corresponding families of moduli spaces are birational over $S$ for any two stability conditions in $U$, and there is a bijection between walls in $N^1(X_R^{[n]}/B)_{\mathbb R,>0}$ inducing contractions, walls in $HH_0([E/\Gamma])$ where the contraction is not regular, and a subset of roots of $R^{ell}$. This predicts that the birational wall structure of the orbifold Hilbert schemes is controlled by deformation walls and elliptic-root-system roots; no resolution is supplied.