My favourite is Oda's Strong Factorization Conjecture:
Can a proper, birational map between smooth toric varieties be factored as a composition of a sequence of smooth toric blow-ups followed by a sequence smooth toric blow-downs?
Note that if you are allowed to intermingle the blow-ups and blow-downs (the weak version) it has been proved. In fact, it was proved for general varieties in characteristic 0 using the toric case:
Torification and Factorization of Birational Maps. Abramovich, Karu, Matsuki, Wlodarczyk.
A conjectural algorithm for computing toric strong factorizations can be found in the following arXiv article:
On Oda's Strong Factorization Conjecture. Da Silva, Karu.