I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.

For a manifold $M$, its cotangent bundle $T^*M$ is naturally a symplectic manifold. Then a section of $\pi:T^*M \rightarrow M$ is Lagrangian if and only if it is a closed $1$-form. Here are my questions:

It is true that two Lagrangian sections are Hamiltonian isotopic iff their difference is exact? If not under what assumption is it true?

Given a Riemannian metric on $M$ that satisfies the real Monge-Ampere equation, it is possible to put a "nice" CY structure on $T^*M$? I know this is true when M is an affine manifold.

Assume they are true, we seem to be able to show the above conjecture by the Hodge theory that de Rham cohomology has harmonic representatives. Any comments and suggestions are welcome!