Howe's duality conjecture for reductive dual pairs

Let $G$) and $G^{\prime}$ be mutual centralizers forming a reductive dual pair inside a symplectic group over a local field $F$. Let $\omega$ be an oscillator representation of the metaplectic group, and let $\mathcal{R}(E)$ denote the set of equivalence classes of continuous irreducible representations of the preimage of a reductive subgroup $E$ that can be realized as quotients of the smooth oscillator representation. Howe's duality conjecture. The set $\mathcal{R}(G\cdot G^{\prime})$ is the graph of a bijection between all of $\mathcal{R}(G)$ and all of $\mathcal{R}(G^{\prime})$. Moreover, an element of $\mathcal{R}(G\cdot G^{\prime})$ occurs as a quotient of $\omega$ in a unique way. This conjecture asserts both multiplicity-free theta correspondence and uniqueness of the corresponding quotient realization for the representations under consideration.