  In [RW19] Rietsch and Williams relate cluster structures and mirror symmetry
for type A Grassmannians Gr(k, n), and use this interaction to construct
Newton-Okounkov bodies and associated toric degenerations. In this article we
define a cluster seed for the Lagrangian Grassmannian, and prove that the
associated Newton-Okounkov body agrees up to unimodular equivalence with a
polytope obtained from the superpotential defined by Pech and Rietsch on the
mirror Orthogonal Grassmannian in [PR13].
