The smooth-normalization conjecture for varieties

Let $X$ be an algebraic or analytic variety. 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. Write $X^{\mathrm{nc}}$ for the normal-crossings locus of $X$. Smooth-normalization conjecture. There is a finite composite of admissible smooth blowings-up

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

preserving $X^{\mathrm{nc}}$, such that $X'$ has smooth normalization. In the analytic case, the morphism is understood over a given relatively compact open subset of $X$. This would provide a controlled partial desingularization while leaving the normal-crossings locus unchanged; the paper presents it as the main conjectural goal and does not establish it in full generality.