Suppose $(M,\omega)$ is a symplectic manifold, $X$ and $X'$ are symplectic submanifolds, and $X$ can be smoothly deformed via symplectic submanifolds to $X'$. Is there a diffeomorphism generated by a Hamiltonian vector field that takes $X$ to $X'$?
Additional necessary conditions and partial answers are welcome. For example, this is true if $X$ is a finite set of points from https://mathoverflow.net/a/62018/50754. Is it true if $X$ is a sphere? Or for example, what if we know that $M$, $X$, $X'$ are Kaehler and the homotopy is via Kaehler submanifolds?