The Euler product conjecture for parabolic rigid Gross $G$-motives

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ satisfy the stated hypotheses, fix a parabolic subgroup $P$, and let $\mathcal{E}$ be an augmented commutative $\mathbb{E}_\infty$-algebra. Let $\mathcal{M}(G)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}}$ be the associated parabolic rigid Gross $G$-motive and let $\mathrm{Frob}$ and $\mathrm{Frob}_x$ denote global and local Frobenius. The parabolic Euler-product conjecture.

$$L_{\mathcal{M}(G),\mathrm{Frob}^{-1},\mathcal{E}}(t):=\mathrm{det}(1-t\mathrm{Frob}^{-1}|\mathrm{H}^*(\overline{X},\mathcal{M}(G)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}})^{-1} =\prod_{x\in X}\mathrm{det}(1-t\mathrm{Frob}_x^{-1}|\mathrm{H}^*(\overline{X},\mathcal{M}(G_x)^{\mathrm{Gross,Parabolic}}_{\mathcal{E}_x})^{-1}.$$

This is presented as a weaker Euler-product consequence of the proposed parabolic Gross-motive theory; the supplied text gives no resolution.