The conjecture on cyclic base change eliminating supercuspidal representations

Let $F_0$ be a local field, let $G$ be a reductive group over $F_0$, let $\pi_0\in\Pi(G/F_0)$ be any supercuspidal representation, and let $\Pi_0\subset\Pi(G/F_0)$ be the packet containing $\pi_0$. Cyclic base-change conjecture. There is a finite sequence of cyclic extensions

$$F_0\subset F_1\subset\dots\subset F_r$$

such that every member of $BC_{F_r/F_0}(\Pi_0)$ contains an Iwahori-fixed vector. The conjecture strengthens the preceding no-pure-incorrigibility claim and is suggested as a consequence of a suitable local Langlands correspondence compatible with parabolic induction; its general status is open.