Adams's conjecture on global theta lifts and A-parameters

Let $F$ be the global field implicit in the automorphic setting, let $W$ be a symplectic space, let $V$ be a quadratic space, and let $\sigma\subset \mathcal{A}_{cusp}({\rm Sp}(W))$ have A-parameter $\Psi$, regarded as an $(\dim W+1)$-dimensional representation of $L_F\times {\rm SL}_2({\mathbb C})$. Assume that the global theta lift of $\sigma$ is a summand in $\mathcal{A}_2({\rm O}(V))$. Adams's conjecture. The global theta lift of $\sigma$ has A-parameter

$$\Psi\oplus S_{\dim V-\dim W-1}.$$

This conjecture predicts the relation between the A-parameters of a cuspidal representation and its square-integrable global theta lift, extending the local unramified calculations described in the paper. Its status is not resolved in the supplied source.