Geck's conjecture relating the preorders on irreducible representations

Let $W$ be a finite Coxeter group with weight function $L$, and let $E,E'\in\text{Irr}(W)$. The preorder $\preceq_L$ is defined using standard representation-theoretic operations, while $\preceq_{\mathcal{LR}}$ is the preorder induced by Kazhdan–Lusztig two-sided cells. Geck's conjecture. One has

$$E\preceq_L E'\quad\Longleftrightarrow\quad E\preceq_{\mathcal{LR}} E'.$$

The conjecture is proved in the equal-parameter case by Geck. The source discusses the unequal-parameter case as the remaining setting, and the paper establishes the relevant type $B$ integer-parameter result.