Quasipositive surfaces and decomposable Lagrangians

We show that a quasipositive surface with disconnected boundary induces map between the knot Floer homology groups of its components preserving transverse invariant. As an application, we this invariant can be used to obstruct decomposable Lagrangian cobordisms arbitrary genus within Weinstein cobordisms. The construction our maps rely on comultiplicativity Along way, also recover various naturality statements for under contact +1 surgery.