The wall-crossing conjecture for iterated Hilbert schemes

Let $G\subseteq\operatorname{SL}_3(\C)$ be abelian, let $A\lhd G$ be a normal subgroup, and write $T=G/A$. Let $Q_G$ be the McKay quiver of $G$, with dimension vector $\ub{d}$, let $\Theta(Q_G,\ub{d})$ be its space of stability conditions, and let $\mathcal{C}_0$, $\mathcal{M}_\gamma(Q_G,\ub{d})$, and $\chi(\mathfrak{w})$ have the meanings specified in the wall-crossing setup. A representation of $G$ is lifted from $T$ if it is obtained by inflating a representation of $T$ along $G\to T$.

Wall-crossing conjecture. There is a path $\gamma\colon[0,1]\to\Theta(Q_G,\ub{d})$ passing through walls $\mathfrak{w}_1,\dots,\mathfrak{w}_m$ satisfying the conditions in the wall-crossing setup such that $\gamma(0)\in\mathcal{C}_0$,

$$\mathcal{M}_{\gamma(1)}(Q_G,\ub{d})\cong T\operatorname{-Hilb}(A\operatorname{-Hilb}(\C^3)),$$

and all nontrivial irreducible representations lifted from $T$ are contained in $\bigcup_{i=1}^m\chi(\mathfrak{w}_i)$. This conjecture proposes a wall-crossing construction of the iterated Hilbert scheme $T\operatorname{-Hilb}(A\operatorname{-Hilb}(\C^3))$ from the $G$-Hilbert scheme.