Outer-automorphism compatibility conjecture for local Langlands reciprocity

Let $G$ be a reductive group, let $\Out_F(G)$ be its group of $F$-rational outer automorphisms, let $\theta\in\Out_F(G)$, and let $\mathfrak w$ be a Whittaker datum. Outer-automorphism compatibility conjecture. For every relevant irreducible representation $\dot\pi$,

$$\operatorname{rec}_{\theta(\mathfrak w)}\bigl(\theta(\dot\pi)\bigr)=\theta\bigl(\operatorname{rec}_{\mathfrak w}(\dot\pi)\bigr).$$

This predicts equivariance of the refined local Langlands correspondence, including the possibility that outer automorphisms move representations between rigid inner twists.