Levi-subgroup compatibility conjecture for local Langlands correspondences

Let $G$ be a reductive group and let $\dot\pi$ be supercuspidal. Levi-subgroup compatibility conjecture. Every Levi subgroup of $G$ satisfies the cuspidality criterion and satisfies unramified-twist and outer-automorphism compatibility for $\dot\pi$. These compatibilities are intended to ensure that the local Langlands constructions agree with the corresponding Bernstein-block structures.