Minimal-valuation local semistable reduction conjecture for F-isocrystals

Let $X$ be a smooth irreducible $k$-variety, let $\overline{X}$ be a partial compactification of $X$, and let $\mathcal{E}$ be an $F$-isocrystal on $X$ overconvergent along $\overline{X}\setminus X$. A minimal valuation on $X$ is a valuation $v$ on $k(X)$ over $k$ such that

$$\operatorname{height}(v)=1\quad\text{and}\quad \operatorname{trdeg}(\kappa_v/k)=0.$$

Minimal-valuation local semistable reduction conjecture. $\mathcal{E}$ admits local semistable reduction at any minimal valuation $v$ on $k(X)$ centered on $\overline{X}$.