Fourier-support conjecture for residual representations

Let $\tau=\tau_1\boxplus\cdots\boxplus\tau_r$ and let $\sigma\in{\mathcal {A}}_{\mathrm{cusp}}(H_m)$ have an $H_m$-relevant generic global Arthur parameter. Let $\mathcal E_{\tau\otimes\sigma}$ be the residual representation defined from the Eisenstein series in the source, and let $\mathfrak p(\mathcal E_{\tau\otimes\sigma})$ be the set of relevant partitions supporting a nonzero Fourier coefficient. Define

\underline p_{\tau;\sigma,H_m^*}=\begin{cases}[(a+\mathfrak m-1)(a+1)]&\text{if }H_m^*=\operatorname{SO}_{2m},\\[(a+\mathfrak m)(a)]&\text{otherwise.}\end{cases}

Fourier-support conjecture. Every $\underline p\in\mathfrak p(\mathcal E_{\tau\otimes\sigma})$ satisfies $\underline p\leq\underline p_{\tau;\sigma,H_m^*}$; if $H_m=H_m^*$ is $F$-quasisplit, there exists a cuspidal $\sigma$ with a nonzero Whittaker-Fourier coefficient for which $\underline p_{\tau;\sigma,H_m^*}\in\mathfrak p(\mathcal E_{\tau\otimes\sigma})$. The conjecture describes the maximal Fourier support of the residual representation and, in the quasisplit case, predicts that the bound is attained. No resolution is supplied in the excerpt.