Unipotent-orbit conjecture for theta representations of metaplectic symplectic groups

Let $r$ be odd and let $r<2n$. Write $2n=a(n,r)r+b(n,r)$ with $a(n,r)$ and $b(n,r)$ nonnegative and $0\leq b(n,r)\leq r-1$. For a partition of $2n$, let its $Sp$ collapse be the greatest symplectic partition smaller than it, where a symplectic partition is one in which every odd part has even multiplicity. Let ${\mathcal O}(\Theta_{2n}^{(r)})$ denote the largest unipotent orbit supporting a nonzero coefficient of the representation $\Theta_{2n}^{(r)}$. Unipotent-orbit conjecture. If $r<2n$, then ${\mathcal O}(\Theta_{2n}^{(r)})$ consists of the partition given by the $Sp$ collapse of $(r^{a(n,r)}b(n,r))$; if $r>2n$, then $\Theta_{2n}^{(r)}$ is generic. This conjecture predicts the unipotent orbit attached to the theta representation and, in the range where $r<2n$, gives a precise partition-theoretic description; the source provides no resolution status.