Diffusion conjecture for random Hamiltonian diffeomorphisms

Let $(M,\omega)$ be a closed symplectic manifold, let $p \in M$ be arbitrary, and let

$$ev_p: \operatorname{Ham}(M,\omega) \to M$$

be the evaluation map at $p$. A law-defining datum is denoted by $\mathcal{D}$, and $\mu_{\operatorname{Ham}}^{\mathcal{D}}$ is the associated probability measure on $\operatorname{Ham}(M,\omega)$.

Diffusion conjecture. For every $\varepsilon>0$, there exists a law-defining datum $\mathcal{D}$ such that the push-forward measure $(ev_p)_*\mu_{\operatorname{Ham}}^{\mathcal{D}}$ and the uniform measure on $M$ are $\varepsilon$-close in a suitable metric, for example total variation or Wasserstein distance.

This conjecture formalizes the observed diffusion of points under random Hamiltonian diffeomorphisms: suitable random Hamiltonians should distribute the image of any fixed point nearly uniformly over the manifold. The supplied text does not state a resolution, so the conjecture remains open.