Conjecture on contragredient invariance for distinguished representations of complex reductive groups

Let $(G,H)$ be a real symmetric pair, where $G$ is a complex connected reductive group. Let $\theta$ be the involution defining $H$, so that $H=G^{\theta}$, and let $\pi\in\mathcal{SAF}_{\mathrm{Irr}}(G)$ be $H$-distinguished, meaning that $(\pi^{*})^{H}\neq 0$. Contragredient-twist conjecture. Then

$$\widetilde{\pi}\simeq\pi^{\theta}.$$

The condition is necessary for $H$-distinction but is not sufficient in general, as shown by complex symmetric pairs with non-distinguished representations satisfying the same isomorphism. The conjecture asserts necessity for all complex connected reductive groups.