Exhaustion conjecture for rank-$k$ representations of finite symplectic groups

Assume that $\dim(U)=k<n$. Let $Sp$ be the finite symplectic group, let $O_{\beta+}$ and $O_{\beta-}$ correspond to the two isomorphism classes of inner products of rank $k$, and let $\eta$ be the eta correspondence embedding the irreducible representations of these orthogonal groups into rank-$k$ representations of $Sp$. Exhaustion conjecture. The rank-$k$ irreducible representations of $Sp$ are exactly those obtained from the two eta correspondences:

$$Irr(Sp)_k=\eta(Irr(O_{\beta+}))\sqcup\eta(Irr(O_{\beta-})).$$

The conjecture asserts that the eta correspondence constructed in the paper produces all rank-$k$ representations; the source presents it as open and gives no resolution.