Let $(U,\omega),(V,\rho)$ be symplectic vector spaces. We call a relation $U \to V$ a Lagrangian relation (also Lagrangian correspondence) if it is a Lagrangian subspace of $\overline U \oplus V$, where $\overline U$ is the conjugate symplectic vector space $(U,-\omega)$.
Lagrangian relations have the property that the composite (as relations) of two Lagrangian relations is again a Lagrangian relation. As any Lagrangian subspace of some space $U$ can be considered a Lagrangian relation $0 \to V$, this implies that Lagrangian relations map Lagrangian subspaces to Lagrangian subspaces. Does the converse hold?
That is, if a relation maps Lagrangian subspaces to Lagrangian subspaces, is it a Lagrangian relation?