The local relative p-adic monodromy conjecture around rank-1 points

Let $X$ be a smooth rigid-analytic space over a $p$-adic field $K$, and let $\mathbb{L}$ be a de Rham $\mathbb{Q}_p$-local system on $X$. A point $x\in X$ has rank $1$ if it is a rank-$1$ point of the adic space. Local relative p-adic monodromy conjecture around rank-1 points. Every rank-$1$ point $x\in X$ admits an open neighborhood $V\subset X$ and a finite étale covering $V'\to V$ such that $\mathbb{L}|_{V'}$ is log-crystalline at all classical points of $V'$. The paper proves this conjecture equivalent to local constancy of the Newton polygon function around rank-$1$ points, but does not establish either condition in general.