The MSV sheaf conjecture for Gorenstein toroidal varieties

Let $X$ be a variety with only Gorenstein toroidal singularities. Let $\mathcal{MSV}(X)$ denote the proposed sheaf of string-differential forms, locally modeled on the product of the corresponding sheaves for an open ball and for a singularity. Its $L[0]=0$ component carries a grading and a differential generalizing the de Rham differential, and its hypercohomology is the associated string cohomology.

MSV sheaf conjecture. For every variety $X$ with only Gorenstein toroidal singularities, there exists a local sheaf $\mathcal{MSV}(X)$ locally isomorphic to the product of $\mathcal{MSV}(\operatorname{open\ ball})$ and $\mathcal{MSV}(\operatorname{singularity})$. The construction depends on the parameters $g_n$ and perhaps on other structures yet to be determined. The sheaf has the structure of a sheaf of conformal vertex algebras, with an $N=2$ structure if $X$ is Calabi--Yau. The hypercohomology of its $L[0]=0$ complex has dimensions prescribed by the cited results and possesses a pure Hodge structure if $X$ is projective.

The conjecture aims to construct a generalized chiral de Rham-type sheaf on varieties with Gorenstein toroidal singularities and recover the expected string-theoretic and Hodge-theoretic invariants. The source calls the definition provisional and says that more work is necessary.