The largest-part conjecture for wave-front sets and local descents

Let $G_n^*$ be the relevant orthogonal group, let $\boldsymbol{p}=[p_1p_2\cdots p_r]$ be the unique maximal partition in the wave-front set ${\mathfrak p}^m(\pi)$, and let $\pi\in\Pi_\phi(G_n^*)$ have generic local $L$-parameter $\phi$. Let $\ell_0=\ell_0(\pi)$ denote the first occurrence index in the local descents.

Largest-part conjecture. The largest part of the partition is

$$p_1=2\ell_0+1.$$

This conjecture predicts that the largest Jordan-block size in the algebraic wave-front set is determined by the first local-descent occurrence. The preceding discussion notes that uniqueness of the maximal partition is itself generally expected beyond the tempered case; the supplied text gives no resolution status for this conjecture.