Cabanes–Enguehard conjecture on d-cuspidal pairs

Prove that, for a finite reductive group G and a fixed integer d, all d-cuspidal pairs (L, θ) that occur under a given irreducible character χ of G (i.e., for which χ has nonzero inner product with Lusztig induction R_L^G(θ)) are G-conjugate.

Background

Within Lusztig’s d-Harish–Chandra theory for finite groups of Lie type, characters of G can be related to d-cuspidal pairs (L, θ) where L is a minimal d-split Levi subgroup and θ is d-cuspidal. The uniqueness (up to G-conjugacy) of such pairs attached to a given irreducible character is a natural structural expectation.

The paper recalls the Cabanes–Enguehard conjecture asserting this uniqueness and notes that, if it holds, it streamlines arguments relating values of characters on picky elements to local data. The authors also mention known cases (e.g., when the parameter s is an ℓ′-element) where the conjecture has been verified, but the general case remains open.