The coincidence conjecture for smooth and algebraic theta correspondences

Let $G$ and $G'$ be a reductive dual pair, let $\widetilde{G}$ and $\widetilde{G'}$ be their corresponding covering groups, and let $\omega_{G,G'}$ be the oscillator representation. Define the smooth and algebraic theta-representation sets by

$$\mathscr{R}^{\infty}(\widetilde{G}\times \widetilde{G'},\omega_{G,G'})\subseteq \mathscr{R}^{\mathrm{alg}}(\widetilde{G}\times \widetilde{G'},\omega_{G,G'}),$$

and similarly for each member of the dual pair, as in the source. Coincidence conjecture. The inclusions

$$\mathscr{R}^{\infty}(\widetilde{G},\omega_{G,G'})\subset \mathscr{R}^{\mathrm{alg}}(\widetilde{G},\omega_{G,G'}),$$ $$\mathscr{R}^{\infty}(\widetilde{G'},\omega_{G,G'})\subset \mathscr{R}^{\mathrm{alg}}(\widetilde{G'},\omega_{G,G'}),$$

and

$$\mathscr{R}^{\infty}(\widetilde{G}\times \widetilde{G'},\omega_{G,G'})\subset \mathscr{R}^{\mathrm{alg}}(\widetilde{G}\times \widetilde{G'},\omega_{G,G'})$$

are all equalities. The conjecture predicts that the smooth and algebraic theta correspondences coincide; the algebraic correspondence is known to be the graph of a bijection, while equality of the smooth and algebraic representation sets is the asserted unresolved coincidence.