The cut-by-curves conjecture for overconvergent F-isocrystals

Let $X$ be a variety, let $Y$ be a boundary or compactification datum, and let $C$ denote a curve contained in the indicated variety. Write $\operatorname{\mathbf{F-Isoc}}(X)$ and $\operatorname{\mathbf{F-Isoc}}(X,Y)$ for the convergent categories with and without the boundary structure, and $\operatorname{\mathbf{F-Isoc}}^\dagger(X)$ for the overconvergent category. Cut-by-curves conjecture. An object of $\operatorname{\mathbf{F-Isoc}}(X)$ extends to $\operatorname{\mathbf{F-Isoc}}(X,Y)$ if and only if, for every curve $C \subseteq Y$, its pullback to $\operatorname{\mathbf{F-Isoc}}(C \times_Y X)$ extends to $\operatorname{\mathbf{F-Isoc}}(C \times_Y X,C)$. In particular, an object of $\operatorname{\mathbf{F-Isoc}}(X)$ extends to $\operatorname{\mathbf{F-Isoc}}^\dagger(X)$ if and only if, for every curve $C \subseteq X$, its pullback to $\operatorname{\mathbf{F-Isoc}}(C)$ extends to $\operatorname{\mathbf{F-Isoc}}^\dagger(C)$. This is known for unit-root objects, while weaker results require the underlying connection to extend to a strict neighborhood.