The generic-degree conjecture for decomposition numbers

Let $W$ be a complex reflection group. Let $\theta:q\mapsto\xi$ be a specialization of ${\mathcal H}_q(W)$, and let $(d_{\chi,\phi}^{\theta})_{\chi\in\operatorname{Irr}(W),\,\phi\in\operatorname{Irr}({\mathcal H}_\xi)}$ be the corresponding decomposition matrix. For each $\phi\in\operatorname{Irr}({\mathcal H}_\xi)$, the generic-degree conjecture asserts that

$$\theta\left(\sum_{\chi\in\operatorname{Irr}(W)}\frac{d_{\chi,\phi}^{\theta}}{s_\chi}\right)\neq 0.$$

Equivalently, using the notation of the source's Lemma $\mathrm{GR}$, $\sum_{\chi\in\operatorname{Irr}(W)}d_{\chi,\phi}^{\theta}D_\chi^P$ has as many factors of the form $(q-\xi)$ as $P(y)$. The conjecture was checked in all cases presented in the paper and was proved for type $A$ and all exceptional Weyl groups, so it is not open in those cases; its general status is not resolved by the supplied text.