The strong factorization conjecture for smooth complete varieties

Let $X$ and $Y$ be smooth complete varieties over a field of characteristic zero, and let $f:X\dashrightarrow Y$ be a birational map. A factorization of $f$ consists of a succession of blow-ups at smooth centers followed by a succession of blow-downs at smooth centers. Strong factorization conjecture. Any such birational map $f$ can be factored in this way. The weak factorization theorem establishes a factorization allowing blow-ups and blow-downs in an interleaved order; requiring all blow-ups to precede all blow-downs remains open.