Bessel-module conjecture for relevant inner forms of unitary groups

Let $G_n$ be a unitary group of $F$-rank ${\mathfrak r}$ with $2{\mathfrak r}< {\mathfrak n}$, and let $G_n^*$ be its $F$-quasisplit pure inner form. Define

$$\underline{p}_{\mathfrak r}=\left[(2{\mathfrak r}+1)1^{{\mathfrak n}-2{\mathfrak r}-1}\right].$$

For a representation $\pi_0$ of $G_n$, write ${\mathcal F}^{{\mathcal O}_{\mathfrak r}}(\pi_0)$ for its ${\mathfrak r}$-th Bessel module associated to this partition. Let $\widetilde{\Phi}_2(G_n^*)_{G_n}$ be the set of $G_n$-relevant generic global Arthur parameters. Bessel-module conjecture. For every $\phi\in\widetilde{\Phi}_2(G_n^*)_{G_n}$, the packet $\widetilde{\Pi}_\phi(G_n)$ contains a member $\pi_0$ such that

$${\mathcal F}^{{\mathcal O}_{\mathfrak r}}(\pi_0)\neq 0.$$

The conjecture extends the known quasisplit case to non-quasisplit unitary groups and is proposed as an input for central-value non-vanishing results. Its general status is not resolved in the supplied text.