The fundamental conjecture for overconvergent isocrystals on partial Frobenius stacks

Let $I$ be an index set, let $X_i$ be the varieties equipped with partial Frobenius actions, and let $X/\Phi$ be the associated partial Frobenius stack. Write $\pi^{\mathrm{Isoc},\dagger}_1(X_i)$ and $\pi^{\mathrm{Isoc},\dagger}_1(X/\Phi)$ for the corresponding overconvergent-isocrystal Tannakian fundamental groups. The fundamental conjecture. The category of $\overline{\mathbb{Q}}_p$-representations of

$$\prod_{i=1}^I \pi^{\mathrm{Isoc},\dagger}_1(X_i)$$

is equivalent to the category of $\overline{\mathbb{Q}}_p$-representations of $\pi^{\mathrm{Isoc},\dagger}_1(X/\Phi)$, and the latter is equivalent to the category of $\overline{\mathbb{Q}}_p$-overconvergent isocrystals over $X/\Phi$. This is the proposed overconvergent, non-étale analogue of Drinfeld's lemma; no resolution is supplied in the text.