The log uniformization conjecture for quasi-excellent schemes

Let $X$ be an integral quasi-excellent scheme and let $\lambda\colon S\to X$ be a valuation. The valuation $\lambda$ is log uniformizable if, for every proper closed subset $D\subsetneq X$, there is a cofinal family of blowings up

$$f_i\colon X_i=\operatorname{Bl}_{V_i}(X)\to X$$

such that each $f_i$ is a log desingularization of $(X,D)$ along $\lambda$.

Log uniformization conjecture. Every valuation on a quasi-excellent integral scheme $X$ is log uniformizable.

This is the local form of the desingularization problem, strengthened to control a boundary and the exceptional locus. The source identifies dimension three as the deepest established case and leaves the general positive-characteristic problem open.