Factorization conjecture for arithmetic automorphic periods of unitary groups

Let $F$ be a CM field, let $\Sigma$ be a CM type, and let $F^+$ be the maximal totally real subfield of $F$. For each signature $I:\Sigma\to\{0,1,\ldots,n\}$, let $U_I$ be the corresponding unitary group and let $P^{(I)}(\Pi)$ be the Petersson period of an algebraic automorphic form in the bottom cohomological degree of its similitude Shimura variety. Let $E(\Pi)$ be the compositum of the rationality fields arising from these periods. Factorization conjecture. There exist nonzero complex numbers $P^{(s)}(\Pi,\sigma)$ for every $0\leq s\leq n$ and $\sigma\in\Sigma$ such that, for every $I=(I(\sigma))_{\sigma\in\Sigma}\in\{0,1,\ldots,n\}^{\Sigma}$,

$$P^{(I)}(\Pi)\sim_{E(\Pi)}\prod_{\sigma\in\Sigma}P^{(I(\sigma))}(\Pi,\sigma).$$

This is the unitary-group generalization of Shimura's period factorization conjecture and is the main conjecture addressed by the paper. The supplied text does not state whether it was proved, so its database status remains open.