Distinguished representations and the moment map conjecture

Let $\pi$ be an irreducible representation of $G$ with $\pi^I\ne0$, and let $(t,n,\chi)$ be its Deligne–Langlands parameter. Let $Z(\pi)$ denote the Aubert–Zelevinsky dual, let $Z(\pi)^\vee$ be its smooth dual, and set

$$a=(t,\sqrt{q_r})\in G^\vee\times\mathbb{C}^\times.$$

Using a Killing form to identify $\mathfrak{g}^\vee$ with $\mathfrak{g}^{\vee *}$, regard $n$ as an element of $\mathfrak{g}^{\vee *}$. Let $M^\vee$ be the hyperspherical dual with moment map

$$\mu:M^\vee\to\mathfrak{g}^{\vee *}.$$

Distinguished-representation moment map conjecture. If $Z(\pi)^\vee$ is $X$-distinguished, then

$$(M^\vee)^a\ne0$$

and $n\in(\mathfrak{g}^{\vee *})^a$ lies in the image of

$$\mu:(M^\vee)^a\to(\mathfrak{g}^{\vee *})^a.$$

This predicts a geometric necessary condition on the Deligne–Langlands parameter of a distinguished representation; the source gives no resolution.