Deligne-style algebraicity conjecture for GSp4 times GL2 L-values

Let $0\pi$ be a cuspidal automorphic representation of $0\operatorname{GSp}_4$ of weight $(k_1,k_2)$, and let $0\sigma$ be a cuspidal automorphic representation of $0\operatorname{GL}_2$ of weight $0\ell$. Let $w$ be the motivic weight of the tensor-product motive associated with $0\pi\times\sigma$, and let $m$ be the \sum of the four smallest Hodge numbers of $M(\pi)\otimes M(\sigma)$. Algebraicity conjecture. There \exists a period $0\Omega(\pi\times\sigma)\in\mathbf{C}^{\times}$ such that, for every $0j\in\mathbf{Z}$ for which $0s=-\tfrac{w}{2}+j$ is critical,

$$\frac{L\left(\pi\times\sigma,-\tfrac{w}{2}+j\right)}{(-2\pi i)^{4j-m}\Omega(\pi\times\sigma)}\in\overline{\mathbf{Q}}.$$

Moreover, for every Dirichlet character $0\chi$,

$$\Omega(\pi\times\sigma\times\chi)=\Omega(\pi\times\sigma)\pmod{\overline{\mathbf{Q}}^{\times}}.$$

This is a period-normalized algebraicity prediction for all critical values, including compatibility with Dirichlet twists; the paper proves some algebraicity results in the relevant critical regions, while the full conjectural period statement remains open.