The vanishing-cycle conjecture for stable framed objects in 3CY categories

Let $\mathcal C$ be a $3CY$ triangulated category and let $\mathcal M^{sfr}$ be the moduli space of stable framed objects. Vanishing-cycle conjecture. There is a formal manifold $\widehat{\mathcal M}^{sfr}$ and a formal function $W\in\widehat{\mathcal O}(\mathcal M^{sfr})$ such that $\mathcal M^{sfr}$ is the set of critical points of $W$, and, for every $i\geq0$,

$$H^i(\mathcal M^{sfr},\phi_W)$$

with coefficients in the vanishing-cycle sheaf $\phi_W(\mathbb Z_{\mathcal M^{sfr}})$ carries a pure Hodge structure of weight $0$ together with the Lefschetz decomposition. The statement is presented as a conjecture for a wide class of triangulated categories, particularly quivers with potential; no resolution is supplied in the source.