  We present a complete classification and the construction of
$\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the
principal series representations, associated to the contact projective geometry
on $\mathbb{RP}^{2n+1}$ and induced from the irreducible
$\mathrm{Mp}(2n,\mathbb{R})$-submodules of the Segal-Shale-Weil representation
twisted by a one-parameter family of characters. The proof is based on the
classification of homomorphisms of generalized Verma modules for the
Segal-Shale-Weil representation twisted by a one-parameter family of
characters, together with a generalization of the well-known duality between
homomorphisms of generalized Verma modules and equivariant differential
operators in the category of inducing smooth admissible modules.
