Prasad's conjecture on distinction of the Steinberg representation

Let $F$ be a nonarchimedean local field with finite residual field, let $E/F$ be a Galois quadratic extension, and let $G$ be a connected reductive group defined over $F$. Write $G_E=G(E)$ and $G_F=G(F)$. For a smooth representation $c0$ of $G_E$ and a character $c7$ of $G_F$, say that $c0$ is $c7$-distinguished if

$$\operatorname{Hom}_{G_F}(c0,c7)\ne 0.$$

Let $G^{ad}=G/Z$ be the adjoint group, and let $c7_{ad}$ be the Prasad character of $G^{ad}(F)$. Prasad's conjecture. The representation $St_E$ is $c7_{ad}$-distinguished with respect to $G^{ad}(F)$, and

$$\operatorname{Hom}_{G^{ad}(F)}(St_E,c7_{ad})$$

is one-dimensional. Moreover, $St_E$ is not $c7'$-distinguished for any character $c7'$ of $G^{ad}(F)$ distinct from $c7_{ad}$. The conjecture generalizes Prasad's result for $G=\operatorname{GL}_2$; the supplied text does not establish whether it has been resolved.