Equivalence of mirror families constructed by toric degenerations of flag varieties

Batyrev (et. al.) constructed a family of Calabi-Yau varieties using small toric degen-erations of the full flag variety G/B. They conjecture this family to be mirror to genericanticanonical hypersurfaces in G/B. Recently Alexeev and Brion, as a part of their workon toric degenerations of spherical varieties, have constructed many degenerations of G/B.For any such degeneration we construct a family of varieties, which we prove coincides withBatyrevs in the small case. We prove that any two such families are birational, thus provingthat mirror families are independent of the choice of degeneration. The birational mapsinvolved are closely related to Berenstein and Zelevinskys geometric lifting of tropical mapsto maps between totally positive varieties.