The exceptional-locus parameterization conjecture for almost large modules

Let $A$ be a finitely generated basic algebra, module-finite over its prime center $Z$, and let $d$ be the dimension vector of a large $A$-module. Assume that $A$ is homologically smooth, that $Z$ is singular, and that a primitive idempotent $e\in A$ satisfies

$$\operatorname{max}\left\{\operatorname{dim}_{\mathbb{C}}(eW)\mid W\text{ a large }A\text{-module}\right\}=1.$$

Assume further that the isoclasses of large $A$-modules are parameterized by the smooth locus of $\operatorname{Max}Z$. Exceptional-locus parameterization conjecture. The isoclasses of almost large $A$-modules $V$ satisfying $\operatorname{Soc}V=e\operatorname{Soc}V$ are parameterized by the exceptional locus $E$ of a smooth resolution $Y\to\operatorname{Max}Z$. For any fixed path-like set $\mathcal{P}$ of $A$, there is a natural bijection between the irreducible components $E_i$ of $E$ and the distinct subsets $P$ such that $P$ is the $\mathcal{P}$-annihilator of an almost large module $V$ with $\operatorname{Soc}V=e\operatorname{Soc}V$, and, if $P=P_{\ell}$ occurs in a maximal chain, the preceding term $P_{\ell-1}$ is the $\mathcal{P}$-annihilator of a large module. If there exists a sequence of $\mathcal{P}$-annihilators

$$0\subsetneq P_1\subsetneq\cdots\subsetneq P_j\subsetneq\cdots\subsetneq P_{\ell},$$

where $P_j$ corresponds to $E_i$, then the isoclasses of almost large modules $V$ with $\operatorname{Soc}V=e\operatorname{Soc}V$ and $\mathcal{P}$-annihilator $P_{\ell}$ are parameterized by a codimension-$\ell$ quasi-projective subvariety of $E_i$ in $Y$. The statement relates representation-theoretic degenerations of large modules to the exceptional geometry of a resolution of the singular center; the supplied source gives no evidence that it has been proved or disproved.