Cabanes--Enguehard transitivity conjecture for e-pairs

Let ${\mathbf{G}}$ be a finite reductive group with Frobenius map $F$. An $e$-pair is a pair $({\mathbf{L}},\lambda)$ where ${\mathbf{L}}$ is an $e$-split Levi subgroup and $\lambda\in{\rm Irr}({\mathbf{L}}^F)$; define $({\mathbf{L}},\lambda)\leq_e({\mathbf{K}},\kappa)$ when ${\mathbf{L}\leq\mathbf{K}}$ and $\kappa$ is an irreducible constituent of the relevant Deligne--Lusztig induction, and let $\ll_e$ be the transitive closure of $\leq_e$.

Cabanes--Enguehard conjecture. The relation $\leq_e$ is transitive and therefore coincides with $\ll_e$. The conjecture concerns the poset of $e$-pairs and would make the directly defined relation sufficient for the theory of $e$-cuspidal pairs; its resolution status is not specified in the source.