The general local base-change criterion via Tate cohomology

Let $F\supset E$ be a degree $l$ extension of characteristic-zero non-archimedean local fields with residue characteristic $p$, where $p$ and $l$ are distinct primes. Let $\pi$ be a cuspidal representation of $\operatorname{GL}_n(E)$ and let $\rho$ be a cuspidal representation of $\operatorname{GL}_n(F)$ that is $\operatorname{Gal}(F/E)$-equivariant, extended to a representation of $\operatorname{GL}_n(F)\rtimes\operatorname{Gal}(F/E)$. General local base-change conjecture. With this notation, $r_l(\pi)$ and $r_l(\rho)$ are in base change exactly when

$$r_l(\pi)^{(l)}\cong T^0(\rho).$$

This extends the result proved in the simplest level-zero, minimal-maximal-type case to arbitrary level and type; its validity in this generality remains open.