The conjecture on pure incorrigible supercuspidal representations

Let $F$ be a local field, let $G$ be a reductive group over $F$, and call a supercuspidal representation pure incorrigible when it is pure and remains supercuspidal in the base-change packet after every finite sequence of cyclic extensions. No pure-incorrigible conjecture. There are no pure incorrigible supercuspidal representations. This is proposed as a generalization of Henniart's splitting theorem; the source notes that the purity condition excludes certain cuspidal unipotent examples and that the conjecture is open in general.