Relative regularization conjecture for integrable connections on polyannuli

Let $K$ be the coefficient field, let $m$ be a positive integer, and let $\epsilon\in(0,1)$. Let $A_K[\epsilon,1)^m$ denote the corresponding polyannulus, and let $\mathcal{E}$ be an integrable $K$-linear connection on it. Relative regularization conjecture. There exists $\epsilon'\in[\epsilon,1)$ and a finite étale covering

$$Y\to A_K[\epsilon',1)^m$$

such that, for every nonarchimedean field $L$ and every morphism $A_L[\epsilon”,1)\to Y\times_K L$ of adic spaces over $L$, the pullback of $\mathcal{E}$ to $A_L[\epsilon”,1)$ is regular as an object of $\mathcal{C}_L$. More precisely, the covering is eligible, meaning that it is induced by a finite étale ring extension of the bounded subring of $\mathcal{O}(A_K[\epsilon',1)^m)$. This question concerns proving global semistable reduction without reducing to projective space; the paper presents it as a question rather than an established result, and no resolution is supplied here.