Finite-dimensionality of cotangent cohomology for $G$-bundle stacks

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ satisfy the stated hypotheses, and let $\mathcal{E}$ be an augmented commutative $\mathbb{E}_\infty$-algebra. Define the cotangent cochain object of the $G$-bundle stack with coefficients by

$$\mathrm{cot}\mathrm{C}^*(\mathrm{Bundle}_G(X\times_{\mathrm{Spec}\mathbb{F}_p}\mathrm{Spec}\overline{\mathbb{F}}_p),h_{\mathrm{Bundle}_G}^*\mathcal{E}).$$

Finite-dimensionality conjecture. Its cohomology

$$\mathrm{H}^{*}(\mathrm{cot}\mathrm{C}^*(\mathrm{Bundle}_G(X\times_{\mathrm{Spec}\mathbb{F}_p}\mathrm{Spec}\overline{\mathbb{F}}_p),h_{\mathrm{Bundle}_G}^*\mathcal{E}))$$

is finite dimensional over $\mathbb{Q}_p$. The claim is motivated by the infinite-degree cohomology of the stack and the proposed cotangent-complex refinement; no proof or resolution is supplied.