The component formula for intertwining-operator poles

Let $C,D\in\mathbf{Comp}$, and let $\pi(C)$ and $\pi(D)$ be the corresponding irreducible representations. For irreducible representations $\pi,\sigma$, let $\Lambda(\pi,\sigma)$ be the order of the pole at $s=0$ of the meromorphically continued intertwining operator $M_{\pi,\sigma}(s)$. Let $\mathfrak{d}(\pi,\sigma)$ be the sum of the two pole orders and the order of the zero at $s=0$ of the corresponding composition. Write $\operatorname{hom}_\Pi(D,C)$ and $\operatorname{ext}^1_\Pi(D,C)$ for the generic Hom and first Ext dimensions attached to the components.

Intertwining-pole conjecture. One has

$$\Lambda(\pi(C),\pi(D))=\operatorname{hom}_\Pi(D,C),\qquad \mathfrak{d}(\pi(C),\pi(D))=\operatorname{ext}^1_\Pi(D,C).$$

This is the paper's main geometric conjecture: it predicts that pole orders of intertwining operators are controlled by Hom and Ext data of Lusztig's characteristic variety. The supplied text does not state a resolution.