Oda's strong factorization conjecture for smooth fans

Let $\Sigma_1$ and $\Sigma_2$ be smooth fans with the same support. A smooth star subdivision is the subdivision of a fan obtained by inserting the ray through the sum of the primitive generators of the rays of a cone, together with the induced subdivisions of its faces. Oda's strong factorization conjecture. There exists a third fan $\Sigma_3$ which can be obtained from both $\Sigma_1$ and $\Sigma_2$ by sequences of smooth star subdivisions. The conjecture is the combinatorial form of strong factorization for toric birational maps. It is proved in the paper for pairs of smooth fans refined by the braid arrangement fan, while its existence in general remains an open problem.