Depth-zero base change compatibility for unramified extensions

Let $F$ be a local field, let $E/F$ be an unramified extension, and let $\underline{G}$ be a reductive group over $F$. For a point $x$ in the building of $\underline{G}(F)$, let $G_x$ be the associated parahoric subgroup and let $\underline{\mathsf{G}}_x$ be its connected reductive quotient over the residue field $k_F$. Write $\widetilde{G}_x$ for the corresponding parahoric subgroup of $\operatorname{Res}_{E/F}\underline{G}(F)=\underline{G}(E)$. A representation has depth zero if it contains $(G_x,\operatorname{infl}(\sigma))$ for an irreducible cuspidal representation $\sigma$ of $\underline{\mathsf{G}}_x(k_F)$. Depth-zero base change compatibility. Suppose there is a base change lifting $\operatorname{BC}_{E/F}$ taking $L$-packets for $\underline{G}(F)$ to $L$-packets for $\underline{G}(E)$. If $\Pi$ is a depth-zero $L$-packet for $\underline{G}(F)$, $\pi\in\Pi$ contains $(G_x,\operatorname{infl}(\sigma))$, and $\sigma$ is irreducible and cuspidal, then some $\widetilde{\pi}\in\operatorname{BC}_{E/F}(\Pi)$ contains

$$(\widetilde{G}_x,\operatorname{infl}(\operatorname{BC}_{k_E/k_F}(\sigma))).$$

Here $\operatorname{BC}_{k_E/k_F}$ is the base change lifting associated with $k_E/k_F$ and the $k_F$-group $\underline{\mathsf{G}}_x$. This conjecture expresses compatibility between base change for depth-zero representations and base change for the corresponding finite reductive groups. It is known, under additional hypotheses, for unramified quadratic extensions and unitary groups in two or three variables; the general assertion remains open.