The subobject morphism conjecture for overconvergent F-isocrystals

Let $X$ be the fixed variety, and let $\operatorname{\mathbf{F-Isoc}}(X)$ and $\operatorname{\mathbf{F-Isoc}}^\dagger(X)$ denote the categories of convergent and overconvergent $F$-isocrystals on $X$. Let $\mathcal{E}_1,\mathcal{E}_2 \in \operatorname{\mathbf{F-Isoc}}^\dagger(X)$ be irreducible, and let $\mathcal{F}_1,\mathcal{F}_2$ be objects of $\operatorname{\mathbf{F-Isoc}}(X)$ that are subobjects of $\mathcal{E}_1,\mathcal{E}_2$, respectively. Subobject morphism conjecture. For every morphism $\mathcal{F}_1 \to \mathcal{F}_2$ in $\operatorname{\mathbf{F-Isoc}}(X)$, there exists a morphism $\mathcal{E}_1 \to \mathcal{E}_2$ in $\operatorname{\mathbf{F-Isoc}}^\dagger(X)$ such that the induced diagram commutes in $\operatorname{\mathbf{F-Isoc}}(X)$. This optimistic conjecture would control how morphisms between convergent subobjects extend in the overconvergent category. An important special case says that an irreducible overconvergent $F$-isocrystal with constant slope polygon is uniquely determined by the first step of its slope filtration; no counterexample is known.