The analytic local uniformization conjecture for rig-smooth spaces

Let $k$ be a valued field and let $X$ be a rig-smooth $k$-analytic space, meaning that $X$ is smooth at every Zariski-closed (rigid) point. For a formal model $\mathfrak{X}$, a point $y\in X^{\operatorname{ad}}$ is uniformizable if there is a finite separable extension $l/k$ and an admissible blowing up $\mathfrak{X}'\to\mathfrak{X}_l$ such that a preimage of $y$ in $X_l^{\operatorname{ad}}$ specializes to a semistable point of $\mathfrak{X}'$.

Analytic local uniformization conjecture. Every point $x\in X^{\operatorname{ad}}$ is uniformizable.

This is the analytic analogue of Zariski local uniformization. The source notes that finite ground-field extension is necessary even for curves and presents the conjecture as the central local problem for rig-smooth analytic spaces.