Tsuzuki's full-faithfulness conjecture for overconvergent isocrystals

Let $k$ be the perfect field and $\mathcal{S}$ the fixed formal base. Let $X$ be a smooth separated $k$-scheme of finite type, and let $X\hookrightarrow\overline{X}$ be an open immersion with $X$ dense in $\overline{X}$. The categories $I^{\dagger}((X,\overline{X})/\mathcal{S}_K)$ and $I^{\dagger}((X,X)/\mathcal{S}_K)$ are the categories of overconvergent isocrystals on the indicated pairs. Tsuzuki's conjecture. The restriction functor

$$I^{\dagger}((X,\overline{X})/\mathcal{S}_K)\longrightarrow I^{\dagger}((X,X)/\mathcal{S}_K)$$

is fully faithful. This conjecture concerns whether morphisms between overconvergent isocrystals can be detected after restricting from a compactification to the pair $(X,X)$; its resolution is not supplied in the source span.