Prasad's distinction conjecture for Iwahori-spherical discrete series

Let $E/F$ be a quadratic unramified extension of nonarchimedean local fields. Let $\mathbb H$ be a simply connected split simple group over $F$ if $-1$ belongs to the Weyl group of $\mathbb H$, and otherwise the unique quasi-split but nonsplit group over $F$ that splits over $E$. Set $H=\mathbb H(F)$ and $G=\mathbb H(E)$. An Iwahori-spherical discrete series representation $\pi$ of $G$ has an $L$-parameter

$$\phi:W_E\times\operatorname{SL}_2(\mathbb C)\longrightarrow{}^LG$$

which factors through an unramified extension of $E$ on $W_E$. Assume that the group of connected components of the centralizer in $\widehat G$ of the corresponding unipotent element is an elementary abelian $2$-group.

Prasad's conjecture. The representation $\pi$ is $H$-distinguished if and only if both the restriction of $\phi$ to $W_E$ is trivial and $\pi$ is generic. Under the first condition, genericity should not depend on the choice of Whittaker datum. Moreover, when both conditions hold,

$$\dim\operatorname{Hom}_H(\pi,\mathbb C)$$

is the order of the centralizer of $\phi(\operatorname{SL}_2(\mathbb C))$ in $\widehat G$.

This is a proposed Langlands-parameter criterion for distinction in the Galois symmetric space $\mathbb H(E)/\mathbb H(F)$. The component-group hypothesis is known for groups of types $B_n$, $C_n$, and $D_n$, while the formulation is presented as a conjectural prediction rather than a theorem in the stated generality.