Lapid's L-packet invariance conjecture for distinguished representations

Let $\underline{G}$ be a connected algebraic group defined over a local field $F$, and let $\theta:\underline{G}\longrightarrow\underline{G}$ be an involution defined over $F$. Set

$$\underline{H}=\underline{G}^{\theta},\qquad G=\underline{G}(F),\qquad H=\underline{H}(F).$$

Let $\pi$ be either in $\mathcal{SAF}_{\mathrm{Irr}}(G)$ in the Archimedean case, or irreducible and smooth in the non-Archimedean case. Assume that $\pi$ is $H$-distinguished, meaning that $(\pi^{*})^{H}\neq 0$. Lapid's conjecture. The $L$-packet of $\pi$ is invariant under the functor

$$\pi\longmapsto\widetilde{\pi}\circ\theta.$$

This generalizes the conjecture for complex connected reductive groups from the Archimedean setting to local fields and includes both Archimedean and non-Archimedean representations.