Generalized Shahidi conjecture for local ABV-packets

Let $\mathrm G$ be a connected reductive group over $F$ with a quasi-split pure inner form, and let $G=\mathrm G(F)$. Let $\Phi(G)$ be the set of local $L$-parameters, let $\Pi^{\mathrm{ABV}}_\phi$ be the corresponding ABV-packet, and let $\widehat\phi$ be the Pyasetskii involution of $\phi$, with associated geometric nilpotent orbit $\mathcal O_{\widehat\phi}$. Generalized Shahidi conjecture for local ABV-packets. For every $\phi\in\Phi(G)$, (i) for every $\pi\in\Pi^{\mathrm{ABV}}_\phi$ and every $\mathcal O\in\overline{\mathfrak n}^{m}(\pi)$,

$$\mathcal O\leq d_{BV}(\mathcal O_{\widehat\phi}),$$

and (ii) if $G$ is quasi-split, at least one $\pi\in\Pi^{\mathrm{ABV}}_\phi$ satisfies

$$\overline{\mathfrak n}^{m}(\pi)=\{d_{BV}(\mathcal O_{\widehat\phi})\}.$$

This extends the packet-sharpness assertion from local Arthur packets to ABV-packets. It is presented as a conjectural generalization and is not known in full generality.