The minimal-singularities conjecture for partial desingularization

Let $X$ be an algebraic or analytic variety, and let $X^{\mathrm{nc}}$ denote its normal-crossings locus. A smooth blowing-up is a blowing-up with smooth centre, and it is admissible when it satisfies the local normal-crossings and Hilbert–Samuel conditions specified in the source. Minimal-singularities conjecture. There is a finite composite of admissible smooth blowings-up

$$\sigma:X'\to X,$$

preserving $X^{\mathrm{nc}}$, such that $X'$ has only singularities from an explicit finite list, called minimal singularities, and every minimal singularity has smooth normalization. This is the paper’s more concrete proposed approach to the smooth-normalization conjecture: it seeks a finite controlled list of residual singularities, but the general assertion remains open.