The depth-zero restricted-parameter conjecture

Let $G$ be a connected reductive group over $F$ that splits over a tame extension. Let $\pi$ be a depth-zero irreducible representation of $G(F)$, and let $\phi_\pi\in\operatorname{RP}_0$ be its restricted depth-zero parameter as constructed in Lemma~. Depth-zero restricted-parameter conjecture. The restriction of a Langlands parameter for $\pi$ to $\operatorname{I}_F/\operatorname{I}_F^{0+}$ lands in the $G^\vee$-conjugacy class of $\phi_\pi$. This is the depth-zero analogue of the paper's positive-depth compatibility conjecture; the source provides no general resolution.