On a conjecture of Fujino and Sato

We revisit results of Fujino--Sato on complete non-projective $\mathbb Q$-factorial toric varieties and their conjectural factorization by flips. We show that their main results admit short conceptual proofs, avoiding any restriction on the dimension and the Picard number, from the general theory of Cox rings and Mori Dream Spaces, once one organizes small $\mathbb Q$-factorial modifications via the GKZ (secondary fan) decomposition of the moving cone. Moreover, we extend this viewpoint beyond the toric case by proving an analogous statement for complete $\mathbb Q$-factorial weak Mori Dream Spaces: any non-projective such variety admits a divisor $D$ and a $D$-flip to a (projective) Mori Dream Space. Our approach highlights the role of chambers and wall-crossing in the secondary fan as a unifying framework for these constructions.