The weight-graded orbit-closure conjecture

Let $G/P$ and $G/Q$ be the homogeneous spaces under consideration, with $Z_0,\ldots,Z_k$ the $G$-orbits in $G/P\times G/Q$, ordered by decreasing size. Let $j:Z_0\to G/P\times G/Q$ be the open-orbit inclusion, let $\delta_{Z_0,m}$ be its Hodge module, and let $\operatorname{gr}^W_s$ denote the $s$th weight-graded piece. Write $\operatorname{IC}_{\overline{Z}_s,m}$ for the intersection-cohomology Hodge module of $\overline{Z}_s$. Weight-graded orbit-closure conjecture. For $s=0,\ldots,k$,

$$\operatorname{gr}^W_s(j_*\delta_{Z_0,m})=\operatorname{IC}_{\overline{Z}_s,m}\{s\},$$

and all remaining subquotients are zero. This conjecturally identifies the weight filtration of the open-orbit push-forward with the intersection-cohomology Hodge modules of the orbit closures; the source says that techniques from an earlier section may establish it, but records no resolution.