Interlacing-pattern conjecture for discrete series of unitary groups

Let $a_1>a_2>\cdots>a_{n+1}$ and $b_1>b_2>\cdots>b_n$ be Langlands parameters, and let $I(a_.,b_.)$ denote the set of possible interlacing patterns between them. For a relevant pair of unitary groups $G=U(p,q)$ and $H=U(p-1,q)$, let $(\pi_1,\pi_2)$ be the corresponding discrete series representations, and write $\operatorname{Hom}_H(\pi_1,\pi_2)$ for the space of $H$-invariant linear forms.

Interlacing-pattern conjecture. There exists a bijective correspondence between $I(a_.,b_.)$ and discrete series representations $(\pi_1,\pi_2)$ of relevant pairs $(G,H)$ with these Langlands parameters and

$$\operatorname{Hom}_H(\pi_1,\pi_2)\ne 0.$$

Equivalently, if $(a_.,b_.)$ and $(a'_.,b'_.)$ have the same interlacing pattern, then

$$\operatorname{Hom}_H(\pi_1,\pi_2)\ne 0$$

if and only if

$$\operatorname{Hom}_H(\pi'_1,\pi'_2)\ne 0.$$

The claim predicts that the existence of an invariant linear form for the restriction from $U(p,q)$ to $U(p-1,q)$ depends only on the interlacing pattern of the two Langlands parameters, rather than on the particular parameters or representations. The supplied parser gives no evidence that this statement has been resolved.