The cohomological product formula for arithmetic coefficients on $G$-bundle stacks

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ satisfy the stated smoothness, affine, semisimplicity, simple-connectedness, and connected-fiber hypotheses, and let $\mathcal{E}\in D^b_{\mathrm{hol}}(X)$ be an augmented $\mathbb{E}_\infty$-ring. Let $h_{\mathrm{Bundle}_G}$ and $h_x$ be the maps defining the global and local pullbacks of $\mathcal{E}$. The cohomological product-formula conjecture.

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

This extends the preceding uncoefficiented product formula to holonomic arithmetic $\mathcal{D}$-module coefficients; the supplied text gives no resolution.