Depth-zero base change compatibility for tamely ramified extensions

Let $F$ be a local field, let $E/F$ be a cyclic tamely ramified extension, and let $\underline{G}$ be a reductive group over $F$. For a point $x$ in the building of $\underline{G}(F)$, let $G_x$ and $\widetilde{G}_x$ be the corresponding parahoric subgroups, with connected reductive quotients $\underline{\mathsf{G}}_x$ and $\widetilde{\underline{\mathsf{G}}}_x$ over $k_F$. If $\pi$ contains $(G_x,\operatorname{infl}(\sigma))$, where $\sigma$ is an irreducible cuspidal representation of $\underline{\mathsf{G}}_x(k_F)$, Depth-zero base change compatibility for tame ramification. Suppose there is a base change lifting $\operatorname{BC}_{E/F}$ taking $L$-packets for $\underline{G}(F)$ to $L$-packets for $\underline{G}(E)$. For every depth-zero $L$-packet $\Pi$ for $\underline{G}(F)$ and $\pi\in\Pi$ containing $(G_x,\operatorname{infl}(\sigma))$, some $\widetilde{\pi}\in\operatorname{BC}_{E/F}(\Pi)$ contains

$$(\widetilde{G}_x,\operatorname{infl}(\widetilde{\sigma}))$$

for some irreducible representation $\widetilde{\sigma}$ of $\widetilde{\underline{\mathsf{G}}}_x(k_F)$ contained in $\ell(\sigma)$. The assertion proposes a finite-group compatibility even though the reductive quotient need not be obtained by restriction of scalars in the ramified case. The source gives no resolution status for this broader conjecture, so it is recorded as open.