Greenfield–Wallach–Katok conjecture on globally hypoelliptic flows

Let $M$ be a smooth connected compact manifold, and let a smooth flow be a smooth $\mathbb R$ action on $M$. The flow is globally hypoelliptic if its associated leafwise Laplacian is globally hypoelliptic; a flow on a torus is Diophantine when its generating constant vector is Diophantine. Greenfield–Wallach–Katok conjecture. Any globally hypoelliptic $\mathbb R$ action on $M$ is smoothly conjugate to a Diophantine flow on the torus. The conjecture is proved when $M$ is a torus, when $\dim(M)\le 3$, and for homogeneous flows on finite-volume homogeneous spaces, but remains open for general compact manifolds.