Folklore conjecture on the coincidence of smooth and algebraic theta correspondence

Let $\widetilde G$ and $\widetilde G'$ be the groups in a reductive dual pair, let $\omega$ be the associated oscillator representation, and let $\Omega$, $\Omega'$ and $\mathcal{R}$ denote the smooth theta-correspondence sets and relation, while $\Omega^{\mathrm{alg}}$, $\Omega'^{\mathrm{alg}}$ and $\mathcal{R}^{\mathrm{alg}}$ denote their algebraic analogues. Thus

$$\Omega\subset\Omega^{\mathrm{alg}},\qquad \Omega'\subset\Omega'^{\mathrm{alg}},\qquad \mathcal{R}\subset\mathcal{R}^{\mathrm{alg}}.$$

Folklore conjecture on theta correspondence. The inclusions

$$\Omega\subset\Omega^{\mathrm{alg}},\qquad \Omega'\subset\Omega'^{\mathrm{alg}},\qquad \mathcal{R}\subset\mathcal{R}^{\mathrm{alg}}$$

are all equalities.

This conjecture asserts that the smooth and algebraic versions of the theta correspondence coincide. It is presented as a folklore conjecture expected since the publication of Howe's work; the supplied source does not state a resolution.