  For a class of maximally degenerate families of Calabi-Yau hypersurfaces of
complex projective space, we study associated non-Archimedean and tropical
Monge-Amp\`ere equations, taking place on the associated Berkovich space, and
the essential skeleton therein, respectively. For a symmetric measure on the
skeleton, we prove that the tropical equation admits a unique solution, up to
an additive constant. Moreover, the solution to the non-Archimedean equation
can be derived from the tropical solution, and is the restriction of a
continuous semipositive toric metric on projective space. Together with the
work of Yang Li, this implies the weak metric SYZ conjecture on the existence
of special Lagrangian fibrations in our setting.
