Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving

Given a closed, orientable Lagrangian submanifold $L$ in a symplectic manifold $(X,\omega )$, we show that if $L$ is relatively exact then any Hamiltonian diffeomorphism preserving $L$ setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on $L$. Curiously, the proof uses only mod-2 coefficients in its singular and Floer cohomology rings.