Weak Fourier-support conjecture for twisted automorphic descent

Let $\tau$ be the datum in the descent diagram, let $m\leq n$, and define

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

Let $\mathfrak p(\mathcal E_{\tau\otimes\sigma})$ denote the relevant partitions for which the residual representation has a nonzero Fourier coefficient. Weak Fourier-support conjecture. For every integer $m\leq n$, there exist a classical group $H_m$ and $\sigma\in{\mathcal {A}}_{\mathrm{cusp}}(H_m)$ satisfying the conditions of the descent diagram such that

$$\underline p^1_{\tau;\sigma,H_m^*}\in\mathfrak p(\mathcal E_{\tau\otimes\sigma}).$$

This weaker assertion is introduced to support the construction of cuspidal automorphic modules; the excerpt gives no evidence that it has been proved or disproved.