Let $X_{\Sigma'} \to X_{\Sigma}$ be a toric birational morphism between smooth and complete toric varieties induced by a regular subdivision $\Sigma' \leq \Sigma$, i.e. every cone in $\Sigma'$ is contained in a cone in $\Sigma$ and both fans have the same support.
Is it true that there exists a fan $\Sigma'' \leq \Sigma'$ such that $\Sigma''$ is constructed from $\Sigma$ by making a finite number of barycentric subdivisions?
Note: this follows from the following toric version of Oda's strong factorization conjecture:
Conjecture: Let $X$ and $Y$ be smooth, complete toric varieties which are birationally equivalent. Does there exist a third variety $Z$ and birational morphisms $Z \to Y$ and $Z \to X$, which are compositions of blow-ups along closed irreducible subvarieties which are the closures of torus orbits?