Toroidal lifting after modifications

Let $X\to B$ be a morphism as in the theorem, and let $U\subset B$ be an open set over which $X$ is toroidal. Set

$$\Sigma=B\setminus U.$$

A modification by blowings up over $\Sigma$ is a composition of blowings up with smooth centers lying over $\Sigma$.

Toroidal lifting conjecture. There exist modifications $X'\to X$ and $B'\to B$, each a composition of blowings up with smooth centers lying over $\Sigma$, and a lifting $X'\to B'$ which is toroidal.

This conjecture seeks a more controlled version of the toroidal-reduction step, retaining the prescribed toroidal locus and requiring all centers to lie over its complement. The paper presents it as a tempting conjectural strengthening, and does not prove it.