Generalized Shahidi conjecture for dual local L-packets

Let $\mathrm G$ be a connected reductive group and let $G=\mathrm G(F)$. Assume the local Langlands correspondence for $G$. For a local $L$-parameter $\phi$, let $\Pi_\phi$ be its local $L$-packet and define the dual packet

$$\widehat\Pi_\phi:=\{\pi\mid\widehat\pi\in\Pi_\phi\},$$

where $\widehat\pi$ is the Aubert–Zelevinsky dual. Generalized Shahidi conjecture for dual local $L$-packets. For every $\pi\in\widehat\Pi_\phi$ and every nilpotent orbit $\mathcal O\in\overline{\mathfrak n}^{m}(\pi)$,

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

and there exists a $\pi\in\widehat\Pi_\phi$ such that

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

Here $\phi=\phi_{\widehat\pi}$ for the representations under consideration. This modification avoids the potentially nonsingleton upper-bound set in the preceding $L$-packet formulation and remains conjectural in general.