Uniqueness of the maximal Fourier-coefficient partition

Let $G$ be a quasisplit classical group and let $\frak{p}^m(pi)$ be the set of maximal partitions attached to nonzero Fourier coefficients of a local or automorphic representation $pi$.

Uniqueness conjecture. The set $\frak{p}^m(\pi)$ contains only one partition in both the local and automorphic cases.

This assertion concerns the uniqueness of the maximal nilpotent-orbit or partition supporting a nonzero Fourier coefficient. The source attributes it to the local and automorphic literature, but supplies no resolution status.