Trinh–Xue's conjecture on intersections of Hecke-algebra blocks

Let $G$ be a finite reductive group, let $e,d\ge1$ be such that $\Phi_e$ and $\Phi_d$ divide the order polynomial of $G$, and let $({\mathbf L},\lambda)$ and $({\mathbf M},\mu)$ be, respectively, an $e$-cuspidal and a $d$-cuspidal pair. For a complete HC-parametrisation $\Psi=\{\Psi_{{\mathbf L},\lambda}\}$, the corresponding Harish-Chandra series are the images of the irreducible characters of the associated cyclotomic Hecke algebras under the maps $\Psi_{{\mathbf L},\lambda}$. Let $\zeta_e$ and $\zeta_d$ be primitive $e$th and $d$th roots of unity, respectively, and let blocks mean the block partitions of the specialised Hecke algebras.

Trinh–Xue's conjecture. There exists a choice of complete HC-parametrisation $\Psi=\{\Psi_{{\mathbf L},\lambda}\}$ such that, for every such pair $({\mathbf L},\lambda)$ and $({\mathbf M},\mu)$, the intersection

$$\Psi_{{\mathbf L},\lambda}(\operatorname{Irr}({\mathcal H}(W({\mathbf L},\lambda),x)))\cap\Psi_{{\mathbf M},\mu}(\operatorname{Irr}({\mathcal H}(W({\mathbf M},\mu),x)))$$

admits a partition $\bigsqcup B_i$ such that, for every $i$, $\Psi_{{\mathbf L},\lambda}^{-1}(B_i)$ is a block of ${\mathcal H}(W({\mathbf L},\lambda),\zeta_d)$ and $\Psi_{{\mathbf M},\mu}^{-1}(B_i)$ is a block of ${\mathcal H}(W({\mathbf M},\mu),\zeta_e)$. In particular, this defines a bijection between the blocks of the two algebras appearing in this intersection.

The conjecture seeks a common block partition for intersections of Harish-Chandra series associated with different cyclotomic parameters. The source notes that the partitions depend on the choice of complete HC-parametrisation and that no analogous relation is known there in general for $e>1$.