Geometric rigidity conjecture for Diophantine step-two nilflows

A nilflow is a flow generated by a one-parameter subgroup on a nilmanifold; a flow is geometrically rigid when continuous conjugacies within the relevant smooth class are smooth. Consider Diophantine Heisenberg nilflows, or more generally Diophantine nilflows of step $2$. Geometric rigidity conjecture for step-two nilflows. Diophantine Heisenberg, or more generally step $2$, nilflows are $C^\infty$ geometrically rigid. The paper proves corresponding rigidity results for Heisenberg nilflows and certain skew-product flows, motivating this conjecture; its status is not resolved in the supplied text.