The compact analytic semistable covering conjecture

Let $k$ be a valued field and let $X$ be a compact rig-smooth $k$-analytic space. A finite covering is a finite union $X_l=\bigcup_{i=1}^n X_i$, and a semistable formal model of $X_i$ is a semistable formal model over $l$.

Compact covering conjecture. There exists a finite extension $l/k$ and a finite covering

$$X_l=\bigcup_{i=1}^n X_i$$

such that each $X_i$ possesses a semistable formal model $\mathfrak{X}_i$ over $l$.

This is a less local analytic formulation implied by the pointwise local uniformization conjecture. It expresses the existence of finitely many semistable formal charts after finite extension.