The Hodge-module open-orbit equivalence conjecture

Let $G/P$ and $G/Q$ be the two homogeneous spaces in the setup, let $Z_0\subset G/P\times G/Q$ be the open $G$-orbit, and let $j:Z_0\to G/P\times G/Q$ be its inclusion. Let $\delta_{Z_0,m}$ be the corresponding Hodge module, and define its underlying Hodge $\mathcal{D}$-module by

$$\mathcal{T}=\mathsf{G}(j_*\delta_{Z_0,m})\langle\operatorname{dim}(G/P)\rangle.$$

Hodge-module open-orbit equivalence conjecture. The object $\mathcal{T}$ is the kernel of an equivalence

$$D(\mathcal{D}_{G/P,h}\operatorname{-mod})\xrightarrow{\sim}D(\mathcal{D}_{G/Q,h}\operatorname{-mod}).$$

This is the Hodge-module enhancement of the preceding proposed equivalence, but the source supplies no evidence that it has been proved or disproved.