The dense-open line-bundle description of the graded kernel

Let

$$\mathfrak{Z}=T^*G/P\times_{\mathfrak g}T^*G/Q$$

and let $\vec{\operatorname{gr}}(\mathcal{T})$ be the associated graded object of the Hodge-module kernel $\mathcal{T}$. Dense-open line-bundle conjecture. There exists a dense subset $\mathfrak{Z}^o$ of $\mathfrak{Z}$ and a line bundle $\mathcal{L}$ on $\mathfrak{Z}^o$ such that

$$\vec{\operatorname{gr}}(\mathcal{T})\cong R^0f_*\mathcal{L},$$

where $f:\mathfrak{Z}^o\to\mathfrak{Z}$ is the inclusion. This proposes a line-bundle model for the graded kernel over a dense part of the Steinberg-type fiber product; the source gives no resolution.