Module-structure conjecture for perfect bases and affine Grassmannian cohomology

Let $\Lambda$ be a preprojective algebra, let $M$ be a $\Lambda$-module of dimension vector $\nu$, and let $\mathbb G(M[t]/t^n)$ denote the corresponding varieties of $\Lambda[t]$-submodules. For a stable MV cycle $Z$ and an irreducible component $Y\in\operatorname{Irr}\Lambda(\nu)$ with general point $M$, suppose that $b_Z=c_Y$. Then the notation $C[\overline{S_+^\nu\cap S^{0}_-}]$ denotes the coordinate ring appearing in the conjecture, and $\bigoplus_{n\in\mathbb N}H^\bullet(\mathbb G(M[t]/t^n))$ is the graded direct sum of the indicated cohomology groups.

Module-structure conjecture. For every preprojective algebra module $M$ of dimension vector $\nu$, the direct sum

$$\bigoplus_{n\in\mathbb N}H^\bullet(\mathbb G(M[t]/t^n))$$

carries the structure of a $C[\overline{S_+^\nu\cap S^{0}_-}]$-module. Moreover, when $b_Z=c_Y$ and $M$ is a general point of $Y$, the isomorphism

$$\Gamma(Z,\mathcal O(n))\cong H^\bullet(\mathbb G(M[t]/t^n))$$

for all $n\in\mathbb N$ is an isomorphism of $C[\overline{S_+^\nu\cap S^{0}_-}]$-modules.

The conjecture seeks a module-theoretic enhancement of the expected representation-theoretic relationship between MV cycles and quiver Grassmannian cohomology. The supplied text gives no resolution status.