The sharp lower-bound partition conjecture for split special orthogonal groups

Let ${\mathrm {G}}_n$ be an $F$-split ${\mathrm{SO}}_{2n+1}$ or ${\mathrm{SO}}_{2n}$, and let $\frak{p}(\pi)$ denote the partitions attached to a cuspidal automorphic representation $\pi$ of ${\mathrm {G}}_n({\mathbb A})$. Sharp lower-bound partition conjecture. The sharp lower-bound partition $\underline{p}_0^{{\mathrm {G}}_n}$ for $\underline{p}\in\frak{p}(\pi)$, as $\pi$ runs over ${\mathcal {A}}_{\mathrm{cusp}}({\mathrm {G}}_n)$, is

$$\underline{p}_0^{{\mathrm{SO}}_{2n+1}}= \begin{cases} [3^e1^{e+1}]&\text{if }n=2e,\\ [3^{e+1}1^e]&\text{if }n=2e+1, \end{cases}$$

and

$$\underline{p}_0^{{\mathrm{SO}}_{2n}}= \begin{cases} [3^e1^e]&\text{if }n=2e,\\ [53^{e-1}1^e]&\text{if }n=2e+1. \end{cases}$$

The source presents these partitions as a conjectural sharp lower bound for Fourier-coefficient partitions of cuspidal automorphic representations of split special orthogonal groups; no resolution is supplied.