  We prove optimal transport stability (in the sense of Andreasson and the
second author) for reflexive Weyl polytopes: reflexive polytopes which are
convex hulls of an orbit of a Weyl group. When the reflexive Weyl polytope is
Delzant, it follows from work of Li, Andreasson, Hultgren, Jonsson, Mazzon,
McCleerey, that the weak metric SYZ conjecture holds for the Dwork family in
the corresponding toric Fano manifold. In particular, we show that the weak
metric SYZ conjecture holds for centrally symmetric smooth Fano toric
manifolds.
