Howe duality conjecture for the similitude theta correspondence

Let $\mathrm{GSp}^+(V_4)$ and $\mathrm{GSO}(U)$ be the similitude groups in the local Weil representation, and let $\widetilde{\Theta}(\pi^+)$ be the smooth theta lift associated with an irreducible admissible representation $\pi^+$ of $\mathrm{GSp}^+(V_4)$. Define $\Theta(\pi^+)$ to be the irreducible quotient of $\widetilde{\Theta}(\pi^+)$ when it is nonzero, and set $\Theta(\pi^+)=0$ otherwise. Howe duality conjecture. For any irreducible admissible representation $\pi^+$ of $\mathrm{GSp}^+(V_4)$, either $\widetilde{\Theta}(\pi^+)=0$ or it is an admissible finite-length representation of $\mathrm{GSO}(U)$; in the latter case there is a unique $\mathrm{GSO}(U)$-invariant submodule $\widetilde{\Theta}'(\pi^+)$ such that

$$\Theta(\pi^+):=\widetilde{\Theta}(\pi^+)/\widetilde{\Theta}'(\pi^+)$$

is irreducible. Moreover, if $\Theta(\pi_1^+)$ and $\Theta(\pi_2^+)$ are both nonzero and isomorphic for two irreducible admissible representations $\pi_1^+$ and $\pi_2^+$, then $\pi_1^+$ and $\pi_2^+$ are isomorphic. Thus the theta lift is injective on isomorphism classes with nonzero image. This is the local Howe duality assertion for the similitude theta correspondence; the conjecture concerns finite length, irreducibility of the quotient, uniqueness, and injectivity of the correspondence.