Functoriality conjecture for the local Langlands correspondence under automorphisms

Let $A$ act on the rigid inner twists of a connected reductive group $G$, on tempered refined Langlands parameters, and on the set $\Pi_{\mathrm{temp}}$ of tempered representations of rigid inner twists. Suppose that $\dot\pi\in\Pi_{\mathrm{temp}}$ corresponds to $(\phi,\rho)$. Functoriality conjecture. For every $a\in A$, the representation $a\dot\pi$ corresponds to $(a\phi,a\rho)$. Equivalently, if $(G_1,\xi_1,z_1,\pi_1)$ and $(G_2,\xi_2,z_2,\pi_2)$ correspond to $(\phi,\rho)$ and $(a\phi,a\rho)$, respectively, then the naturally constructed isomorphism $b:G_1\to G_2$ identifies $\pi_1$ with $\pi_2$. This asserts naturality of the refined local Langlands correspondence under the automorphism action; the source presents it as an expected compatibility for connected groups.