Centralizer conjecture for uniquely ergodic Heisenberg nilflows

Let $H_3(\mathbb R)$ be the three-dimensional Heisenberg group, let $\Gamma< H_3(\mathbb R)$ be a cocompact lattice, and let

$$M=\Gamma\backslash H_3(\mathbb R).$$

Let $\phi_{\mathbb R}$ be a uniquely ergodic, or Diophantine, Heisenberg nilflow on $M$. The essential measurable centralizer consists of measurable transformations commuting with the flow modulo null sets, while the essential $C^0$ centralizer consists of continuous transformations with the analogous property. Centralizer conjecture for Heisenberg nilflows. The essential measurable centralizer coincides with the essential $C^0$ centralizer. This is proposed in analogy with Ratner's result for horocycle flows; the supplied text does not state whether it has been proved.