Automorphic Deligne conjecture for critical Rankin–Selberg values

Let $K$ be the quadratic imaginary field under consideration. Let $\Pi$ and $\Pi'$ be regular algebraic cuspidal representations of $GL_n(\mathbb{A}_K)$ and $GL_{n'}(\mathbb{A}_K)$, with rationality fields $E(\Pi)$ and $E(\Pi')$. Let $P^{(j)}(\Pi)$ and $P^{(k)}(\Pi')$ be the automorphic periods, and let $sp(j,\Pi;\Pi')$ and $sp(k,\Pi';\Pi)$ be the split indices. Assume $\Pi\times\Pi'$ is critical in the sense that their infinity types have no equality $a_i+b_j=-(\omega(\Pi)+\omega(\Pi'))/2$. Automorphic Deligne conjecture. If $m\in\mathbb{Z}+(n+n')/2$ is critical for $\Pi\times\Pi'$, then

$$L(m,\Pi\times\Pi')\sim_{E(\Pi)E(\Pi');K}(2\pi i)^{nn'm}\prod_{j=0}^{n}P^{(j)}(\Pi)^{sp(j,\Pi;\Pi')}\prod_{k=0}^{n'}P^{(k)}(\Pi')^{sp(k,\Pi';\Pi)}.$$

This is the automorphic analogue of the tensor-product Deligne conjecture, expressing critical Rankin–Selberg values through automorphic periods; the source gives no resolution status for this general statement.