Prasad's multiplicity formula for Galois distinction

Let $E/F$ be a quadratic extension with nontrivial Galois element $\sigma$. Let $G$ be a reductive group over $F$, let $G(E)$ be its base change, and let $G^{\mathrm{op}}$ be the quasi-split $F$-form of $G$ with $G^{\mathrm{op}}(E)\cong G(E)$. For a regular supercuspidal representation $\pi$ of $G(E)$ with Langlands–Vogan parameter $(\phi_\pi,\lambda_\pi)$, write $S_{\phi_\pi}=C_{\widehat G}(\operatorname{Im}\phi_\pi)$. For each $\alpha\in H^1(\operatorname{Gal}(E/F),G(E))$, let $G_\alpha$ be the corresponding inner form, and let $\omega_{G_\alpha(F),E}$ be Prasad's quadratic character. For a lift $\widetilde\phi:W_F\to{}^LG^{\mathrm{op}}$ with $\widetilde\phi|_{W_E}=\phi_\pi$, let $m(\lambda_\pi,\widetilde\phi)$ be the multiplicity of the trivial representation in $\lambda_\pi|_{\pi_0(S_{\widetilde\phi})}$. Prasad's multiplicity formula. One has

$$\sum_{\alpha\in H^1(\operatorname{Gal}(E/F),G(E))}\dim\operatorname{Hom}_{G_\alpha(F)}(\pi,\omega_{G_\alpha(F),E})=\sum_{\widetilde\phi}m(\lambda_\pi,\widetilde\phi),$$

where the right-hand sum is over all such lifts $\widetilde\phi$. This is the regular-supercuspidal form of Prasad's conjecture on Galois distinction and base change; the supplied text does not state whether it has been proved in this generality.