The shtuka cohomology conjecture for proper rigid-analytic varieties

Let $C$ be the completed algebraic closure of a $p$-adic field, let $\mathcal{Y}_{\operatorname{FF}}=\operatorname{Spa}(A_{\inf},A_{\inf})\setminus V([p^{\flat}])$, and let $A$ be the ring of analytic functions on $\mathcal{Y}_{\operatorname{FF}}$. A shtuka of perfect complexes relative to $\operatorname{Spa}\mathbb{Z}_p$ with one leg at $\varphi^{-1}(y_C)$ is a perfect complex $\mathcal{E}$ on $\mathcal{Y}_{\operatorname{FF}}$ equipped with an isomorphism

$$\varphi_{\mathcal{E}}:(\varphi^*\mathcal{E})|_{\mathcal{Y}_{\operatorname{FF}}\setminus\{\varphi^{-1}(y_C)\}}\cong \mathcal{E}|_{\mathcal{Y}_{\operatorname{FF}}\setminus\{\varphi^{-1}(y_C)\}}$$

that is meromorphic at $\varphi^{-1}(y_C)$. The shtuka cohomology conjecture. There exists a cohomology theory $R\Gamma_{\operatorname{sht}}(X/\mathcal{Y}_{\operatorname{FF}})$ for proper rigid-analytic varieties $X$ over $C$, taking values in such shtukas, satisfying: (i) if $X$ is the generic fiber of a proper $p$-adic formal scheme $\mathfrak{X}$ over $\mathcal{O}_C$ with semistable reduction, it is naturally isomorphic to the shtuka associated to $R\Gamma_{A_{\inf}}(\mathfrak{X})\otimes_{A_{\inf}}A$; and (ii) after restriction to $Y_{\operatorname{FF}}$, its cohomology groups are shtukas of vector bundles whose associated admissible modification corresponds, via Fargues' equivalence, to the modification defined by the de Rham lattice. This would provide a direct geometric shtuka-valued cohomology theory for proper rigid-analytic varieties and relate it to $A_{\inf}$-cohomology; the source gives no evidence of a resolution.