Compatibility of cyclic base change with semisimple local Langlands parameters

Let $F' / F$ be an extension, let $F”/F'$ be cyclic, and let $BC_{F”/F'}$ be the conjectural cyclic base-change map on representations of $G(F')$. Let ${\mathcal{L}}^{ss}_{F'}$ and ${\mathcal{L}}^{ss}_{F”}$ denote the semisimple local Langlands parametrizations. Compatibility conjecture. For every $\pi\in\Pi(G/F')$, the semisimple parameter of each base change satisfies

$${\mathcal{L}}^{ss}_{F”}(BC_{F”/F'}(\pi))={\mathcal{L}}^{ss}_{F'}(\pi)|_{W_{F”}}.$$

This predicts that cyclic base change restricts Weil-group parameters as expected. It depends on the existence of the base-change structure and is presented as a conjectural property rather than a theorem.