The Asai local Langlands compatibility conjecture

Let $Fsubset K$ be a quadratic extension of local fields, let $G_n(K)=GL(n,K)$, and let $W'_K$ and $W'_F$ be the Weil–Deligne groups. For a finite-dimensional representation $\rho$ of $W'_K$, let $M_{W'_K}^{W'_F}(\rho)$ be its multiplicative induction to $W'_F$, and define

$$L_F^K(\rho,s)=L\bigl(M_{W'_K}^{W'_F}(\rho),s\bigr).$$

If $\pi$ is an irreducible representation of $G_n(K)$ corresponding to $\rho$ under the local Langlands correspondence, write $L_F^{K,W}(\pi,s)=L_F^K(\rho,s)$. The Asai local Langlands compatibility conjecture. For every generic representation $\pi$ of $G_n(K)$, with corresponding $n$-dimensional Weil–Deligne representation $\rho$ of $W'_K$, one has

$$L_F^K(\pi,s)=L_F^K(\rho,s).$$

The conjecture asserts that the analytically defined Asai $L$-function of a generic representation agrees with the canonical Asai $L$-function obtained from multiplicative induction on the Weil–Deligne side. The source presents this equality as expected; no resolution is supplied there.