The log-smooth modification conjecture for rig-smooth formal schemes

Let $k$ be a complete real-valued field, and let $\mathfrak{X}$ be an admissible formal $\mathcal{O}_k$-scheme whose generic fiber $\mathfrak{X}_\eta$ is rig-smooth. A formal scheme is polystable when it has the polystable local structure considered in the paper.

Log-smooth modification conjecture. There exists a finite extension $l/k$ and an admissible blowing up

$$\mathfrak{X}'\to\mathfrak{X}_l$$

such that $\mathfrak{X}'$ is polystable.

This is the main global conjecture about constructing nice formal models. It strengthens the weaker existence of one semistable or log-smooth model by requiring modification after base extension; the source also explains that log-smooth models can be refined to polystable ones using the result of Abramovich, Lepage, and Temkin.