Generalized Shahidi conjecture for 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

$$\mathrm{UB}(\phi)=\max\{d_{BV}(\mathcal O_{\phi_{\widehat\pi}})\mid \pi\in\Pi_\phi\}.$$

Generalized Shahidi conjecture for local $L$-packets. For every $\pi\in\Pi_\phi$ and every nilpotent orbit $\mathcal O\in\overline{\mathfrak n}^{m}(\pi)$, there is an $\mathcal O'\in\mathrm{UB}(\phi)$ with $\mathcal O\leq\mathcal O'$, and for every $\mathcal O'\in\mathrm{UB}(\phi)$ there is a $\pi\in\Pi_\phi$ such that

$$\overline{\mathfrak n}^{m}(\pi)=\{\mathcal O'\}.$$

This is a proposed extension of the Shahidi conjecture to arbitrary local $L$-packets. Its first part is equivalent to the upper bound conjecture, while the possible nonsingleton nature of $\mathrm{UB}(\phi)$ makes the full assertion difficult and unresolved.