The cohomological product formula for rigid Gross $G$-motives

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ satisfy the stated hypotheses, and let $\mathcal{E}\in\mathbb{D}_{\mathrm{hol}}(X)$ be an augmented commutative $\mathbb{E}_\infty$-algebra. Let $\mathrm{pr}$ and $h_{\mathrm{Bundle}_G}$ define the global coefficient object and let $h_x$ define its local restriction. The rigid Gross $G$-motive product-formula conjecture.

$$\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^*(\mathrm{Bundle}_G(X\times_{\mathrm{Spec}\mathbb{F}_p}\mathrm{Spec}\overline{\mathbb{F}}_p),\mathrm{pr}_*h_{\mathrm{Bundle}_G}^*\mathcal{E})) =\prod_{x\in X}\mathrm{Tr}(\varphi^{-1}|\mathrm{H}^*(\mathrm{Bundle}_G(\{x\}\times_{\mathrm{Spec}\mathbb{F}_p)}\mathrm{Spec}\overline{\mathbb{F}}_p),h_x^*\mathcal{E}_x)).$$

This is the proposed arithmetic $\mathcal{D}$-module version of the Gross $G$-motive product formula; the supplied text gives no resolution.