Włodarczyk's weak factorization conjecture

Let $f:X\dashrightarrow Y$ be a birational map of smooth complete varieties over an algebraically closed field of characteristic zero, which is an isomorphism over an open set $U$. Then $f$ can be factored as

$$X=X_0\dashrightarrow X_1\dashrightarrow\cdots\dashrightarrow X_n=Y,$$

where each $X_i$ is a smooth complete variety and each birational map is a blow-up or blow-down at a smooth center and is an isomorphism over $U$. Moreover, if $X\setminus U$ and $Y\setminus U$ are divisors with normal crossings, then each $D_i:=X_i\setminus U$ is a divisor with normal crossings and each map is a blow-up or blow-down at a smooth center having normal crossings with the components of $D_i$.

Weak factorization conjecture. Every such birational map admits this factorization, with the stated preservation of the open set and normal-crossings boundary.

The conjecture is the central factorization statement developed in the paper and is presented there as proved in characteristic zero. A stronger factorization statement, in which all blow-ups precede all blow-downs, remains open according to the surrounding text.