  We prove that the Gromov width of coadjoint orbits of the symplectic group is
at least equal to the upper bound known from the works of Zoghi and Caviedes.
This establishes the actual Gromov width. Our work relies on a toric
degeneration of a coadjoint orbit to a toric variety. The polytope associated
to this toric variety is a string polytope arising from a string
parametrization of elements of a crystal basis for a certain representation of
the symplectic group.
