Constant-exponent conjecture for regular relative connections

Let $I$ be open, let $X$ be connected and affinoid with a finite Shilov boundary, and let $\mathcal{E}$ be a regular connection of rank $n$. Suppose there are $x\in X$, a closed subinterval $J\subseteq I$ of positive length, and $A\in\mathbb{Z}_p^n$ with $p$-adic non-Liouville differences such that $A$ is an exponent for $\mathcal{E}_x$ on $J$. Constant-exponent conjecture. Then for every $x\in X$, $A$ is an exponent for $\mathcal{E}_x$. This would show that, under the stated non-Liouville hypothesis, an exponent observed on one fiber is constant across a connected affinoid base; the preceding variable-exponent example shows why a connectedness hypothesis is relevant. No resolution is supplied here.