Arthur's theta-lift A-parameter conjecture for unitary groups

Let $E/k$ be the relevant quadratic extension, let ${\rm U}(V)\times {\rm U}(W)$ be a dual pair in the stable range, and let $\pi\subset\mathcal{A}_{\rm cusp}({\rm U}(V))$ have A-parameter $\Psi$, viewed as an $\dim V$-dimensional representation of $L_E\times {\rm SL}_2(\mathbb C)$. Let $\chi_V$, $\chi_W$, and $\psi$ be the data defining the global theta lift. Arthur's theta-lift conjecture. The global theta lift of $\pi$ to ${\rm U}(W)$, when nonzero, is a summand of $\mathcal{A}_2({\rm U}(W))$ with A-parameter

$$\chi_V\cdot\left(\chi_W^{-1}\Psi\oplus S_{\dim W-\dim V}\right).$$

The claim predicts how theta correspondence transforms Arthur parameters and is presented as a natural consequence of the stable-range automorphy of global theta lifts.