(Joint work with J. Palmer)  Suppose that $\phi_t: L \to X, t \in [0,T]$ is a family of Lagrangian immersions flowing under a Maslov flow such as a reverse mean curvature flow, or reverse mean curvature flow coupled to a K\"ahler-Ricci flow as introcuded by Smoczyk.  If $\phi_0$ is weakly unobstructed then so is $\phi_T$ and $HF(\phi_0) \cong HF(\phi_T)$.  This is part of a conjecture of Joyce,  and related to the speaker's Lagrangian minimal model program.