Counterexample: set $U$ to be anything, $V := \text{pt}$, and $\Lambda \subset \overline{U} \oplus \text{pt}$ to be a non-Lagrangian subspace.

Counterexample: set $U$ to be anything, $V := \text{pt}$, and $\Lambda \subset \overline{U} \oplus \text{pt}$ to be a non-Lagrangian subspace.

(But maybe true with some hypotheses, e.g. $\Lambda$ induces an injection on Lagrangian subspaces? Hmm, a linear symplectic analogue of Orlov's theorem...)