The MVW-type innerform conjecture for automorphisms of representations

Let $G$ and $G'$ be inner forms, and let $\phi$ and $\phi'$ be automorphisms arising from the corresponding adjoint groups. Suppose they correspond under the natural isomorphism of the relevant quotients, and let $\pi$ be an irreducible admissible representation of $G(F)$ with a Whittaker model fixed by $\phi$. The MVW-type innerform conjecture. The automorphism $\phi$ acts trivially on the $L$-packet of $G(F)$ containing $\pi$, and the corresponding automorphism $\phi'$ acts trivially on the corresponding $L$-packet of $G'(F)$.