Strong factorization conjecture for birational maps

Work over an algebraically closed field $K$ of characteristic $0$. Let $\phi:X_1\dashrightarrow X_2$ be a birational map between complete nonsingular algebraic varieties, and let $U\subset X_1$ be an open set where $\phi$ is an isomorphism. The weak factorization theorem provides the situation described above.

Strong factorization conjecture. There exists a diagram

$$\begin{array}{rcccl} & & Y & & \\ & \psi_1\swarrow & & \searrow \psi_2 & \\ X_1 & & \stackrel{\phi}{\dashrightarrow} & & X_2 \end{array}$$

where the morphisms $\psi_1$ and $\psi_2$ are composites of blowings up of smooth centers disjoint from $U$.

This strengthens weak factorization by requiring the birational map to be dominated by a common smooth variety through morphisms that are each composites of blowings up, with all centers disjoint from the locus where $\phi$ is already an isomorphism. The supplied source does not state a resolution status for this conjecture.