The monodromy conjecture for relatively exact Lagrangians

Let $M$ be a compact or Liouville symplectic manifold, and let $L\subseteq M$ be a compact, relatively exact Lagrangian. Let $\tilde\phi:L\to L$ be a diffeomorphism extending to a Hamiltonian diffeomorphism $\phi:M\to M$. Monodromy conjecture. The diffeomorphism $\tilde\phi$ is isotopic to the identity. This would strengthen known constraints on Hamiltonian extensions, which act trivially on various cohomological and homotopical invariants of $L$.