It is well known since a few years that a weak version of the metric SYZ conjecture follows from existence and/or structural results about Monge-Ampère equations. For general maximal degenerations of Calabi-Yau manifolds, this principle is formulated in terms of non-Archimedean geometry, but for certain families of hypersurfaces in toric Fano manifolds, we can be a bit more concrete and the weak SYZ conjecture reduces to solvability of a real Monge-Ampère equation on the boundary of a polytope. Curiously, this Monge-Ampère equation is solvable for some families and not solvable for others. I will talk about how solvability can be characterized by a condition from optimal transport theory and present positive as well as negative examples. This is based on joint work with Rolf Andreasson, Thibaut Delcroix, Mattias Jonsson, Enrica Mazzon and Nick McCleerey.