Compatible-splitting conjecture for diagram configuration varieties

Let $D=(C_1,C_2,\ldots,C_r)$ be a diagram, let $\operatorname{Gr}(D)=\operatorname{Gr}(C_1)\times\cdots\times\operatorname{Gr}(C_r)$, and let $\mathcal F_D$ be its configuration variety. Let $u_1,\ldots,u_r$ be Weyl group elements of $\Sigma_N$.

Compatible-splitting conjecture. For any diagram $D$ and any Weyl group elements $u_1,\ldots,u_r$, the pairs

$$\mathcal F_D\subset\operatorname{Gr}(D)$$

and

$$\mathcal F_{u_1,\ldots,u_r}\subset (G/B)^r$$

are compatibly split.

Earlier in the source, compatible splitting is proved under the additional condition that the Weyl group elements form a sequence satisfying the stated weak-order and column conditions. The candidate omits those hypotheses and the parser supplies no resolution, so the unrestricted assertion remains open in this record.