Rationality conjecture for critical L-values of RACARs on GL(3)

Let $\Pi$ be a RACAR of $\mathrm{GL}_3$, let $E$ be its coefficient field, and let $\eta$ be a Dirichlet character with Gauss sum $G(\eta)$. For $j\in\mathbb{Z}_{\geq 0}$, set

$$e_\infty(\Pi,-j)=i^{j-a-1}\Gamma_{\mathbb{C}}(a+1-j)=2(2\pi i)^{j-a-1}\Gamma(a+1-j).$$

Assume $(-j,\eta)\in\operatorname{Crit}^-(\Pi)$. Rationality conjecture. There exists a complex period $\Omega_\Pi^-\in\mathbb{C}^\times$ such that

$$e_\infty(\Pi,-j)\frac{L(\Pi\times\eta,-j)}{\Omega_\Pi^-}\in E[\eta,G(\eta)].$$

This predicts algebraicity of normalized critical values of the standard Rankin–Selberg $L$-function after division by a single period; the supplied text gives no resolution status.