The toroidalization conjecture for morphisms of nonsingular varieties

Let $X$ and $Y$ be nonsingular varieties over an algebraically closed field of characteristic $0$, and let $f:X\longrightarrow Y$ be a dominant morphism. Let $D_Y$ be a simple normal crossing divisor on $Y$ such that

$$D_X=f^{-1}(D_Y)$$

is a simple normal crossing divisor on $X$ containing the singular locus $\operatorname{Sing}(f)$ of $f$. A nonsingular subvariety contained in a simple normal crossing divisor and intersecting it transversally is called a possible center; the blowup along such a center is a possible blowup. Toroidalization conjecture. There exist commutative morphisms

\xymatrix @R=3pc @C=3pc{ X_1 \ar[r]^{f_1} \ar[d]^{\pi_1} & Y_1 \ar[d]^{\pi} \\ X \ar[r]^f & Y }

where $\pi$ and $\pi_1$ are possible blowups for the preimages of $D_Y$ and $D_X$, respectively, such that $f_1$ is toroidal with respect to

$$D_{Y_1}=\pi^{-1}(D_Y),\qquad D_{X_1}=\pi_1^{-1}(D_X).$$

The conjecture seeks a toroidal model of every such dominant morphism after compatible blowups of the source and target, providing a general structure theorem for morphisms of varieties. The supplied text gives no resolution status, so the database status remains open.