The theta-lift transfer conjecture for quaternionic dual pairs

Let $\epsilon=1$, let $\phi$ be a tempered $L$-parameter for $G_0(V)$, and let $\phi'$ be a tempered $L$-parameter satisfying the paper's parameter relation. Let $\mathscr{T}_\psi$ denote the transfer map defined from the local theta correspondence data.

Theta-lift transfer conjecture. If such a parameter $\phi'$ exists, then

$$\theta_\psi(\pi,W)=\mathscr{T}_\psi(\pi).$$

The statement identifies the local theta lift with the transfer map constructed in the paper, under the existence of the corresponding parameter. The supplied excerpt does not indicate that this conjecture has been resolved.