The local theta correspondence conjecture for quaternionic dual pairs

Let $D$ be a division quaternion algebra over $F$, let $V$ be a Hermitian space over $D$, and let $W$ be a skew-Hermitian space over $D$. Let $\psi\colon F\to\mathbb{C}^1$ be a non-trivial character, and let $(z_+,\varphi_+)\in\mathcal{RIT}^\star(V^\#,V)$ and $(z_-,\varphi_-)\in\mathcal{RIT}^\star(W^\#,W)$ be corresponding rigid inner twists. Write $G_0(W)$ for the Zariski connected component of $G(V)$ containing $1$, and assume that $\dim W-\dim V$ is $0$ or $1$. Let $\xi$ be the relevant embedding of $L$-groups, and let $\phi$ and $\phi'$ be tempered $L$-parameters of $G(V)$ and $G_0(W)$ with $\phi=\xi\circ\phi'$.

Local theta correspondence conjecture. If $s\in S_\phi^+$ and $s'\in S_{\phi'}^+$ are associated via $\xi$, and $\pi\in\Pi_\phi(G(V))$, then $\theta_\psi(\pi,W)$ has $L$-parameter $\phi'$ and

$$\iota_\phi[\mathfrak{w}_+,z_+,\varphi_+](\pi)(s)=\overline{\iota_{\phi'}[\mathfrak{w}_-,z_-,\varphi_-](\theta_\psi(\pi,W))(s')}.$$

This conjecture describes the quaternionic local theta correspondence in terms of Langlands parameters and the character relations defining the local Langlands packets. Non-vanishing in the stated range is known, but the asserted parameter identity and character formula are the conjectural part.