Langlands program for $p$-adic coefficients

Let $p$ be a prime, let $k$ be a finite field with $q=p^s$ elements, and let $X$ be a smooth proper geometrically connected curve over $\operatorname{Spec}(k)$ with function field $\mathcal{K}$. Let $\mathcal{A}_r$ be the set of isomorphism classes of irreducible cuspidal automorphic representations of $\operatorname{GL}_r(\mathbb{A}_{\mathcal{K}})$ with finite-order central character. Let $\mathcal{I}_r$ be the direct limit, over dense open subschemes $U\subset X$ and finite extensions of $K_0=W(k)\otimes\mathbb{Q}$ in $\overline{\mathbb{Q}}_p$, of the isomorphism classes of absolutely irreducible overconvergent $F$-isocrystals of rank $r$ with finite determinant. For an object, write $U$ for the common set of unramified places, and compare Frobenius eigenvalues with Hecke eigenvalues at $x\in|U|$. Langlands program for $p$-adic coefficients. There are unique maps

$$\pi_{\bullet}\colon\mathcal{I}_r\rightarrow\mathcal{A}_r,\qquad E_{\bullet}\colon\mathcal{A}_r\rightarrow\mathcal{I}_r$$

such that, for every $E\in\mathcal{I}_r$, the Frobenius eigenvalues of $E$ at every $x\in|U|$ coincide with the Hecke eigenvalues of $\pi_E$, and, for every $\pi\in\mathcal{A}_r$, the Frobenius eigenvalues of $E_\pi$ at every $x\in|U|$ coincide with the Hecke eigenvalues of $\pi$. Moreover,

$$E_{\bullet}\circ\pi_{\bullet}=\operatorname{id}_{\mathcal{I}},\qquad \pi_{\bullet}\circ E_{\bullet}=\operatorname{id}_{\mathcal{A}}.$$

This is the proposed $p$-adic-coefficient form of the Langlands correspondence for curves, matching unramified local data in both directions. The supplied text gives no resolution, so the conjecture is recorded as open.