Howe duality conjecture for local theta correspondence

Let $U$ and $V$ be members of a reductive dual pair, let $\operatorname{\bar{G}}(U)$ and $\operatorname{\bar{G}}(V)$ be the corresponding covering groups, and let $\omega_{U,V}$ be a genuine smooth oscillator representation of $\operatorname{\bar{J}}(U,V)$. Define

$$\mathcal R_{\omega_{U,V}}(U):=\{\pi\in\operatorname{Irr}(\operatorname{\bar{G}}(U))\mid \operatorname{Hom}_{\operatorname{\bar{G}}(U)}(\omega_{U,V},\pi)\neq 0\},$$

and define $\mathcal R_{\omega_{U,V}}(V)$ similarly. Also define

$$\mathcal R_{\omega_{U,V}}(U,V):=\{(\pi,\pi')\in\operatorname{Irr}(\operatorname{\bar{G}}(U))\times\operatorname{Irr}(\operatorname{\bar{G}}(V))\mid \operatorname{Hom}_{\operatorname{\bar{G}}(U)\times\operatorname{\bar{G}}(V)}(\omega_{U,V},\pi\otimes\pi')\neq 0\}.$$

Howe duality conjecture. The set $\mathcal R_{\omega_{U,V}}(U,V)$ is the graph of a bijection between $\mathcal R_{\omega_{U,V}}(U)$ and $\mathcal R_{\omega_{U,V}}(V)$.

This conjecture asserts multiplicity-free, bijective local theta correspondence for the representations occurring in the oscillator representation. Its status is not specified in the supplied text.