Heisenberg-group equidistribution conjecture for Diophantine parameters

Let $\mathbb{H}(\mathbb{R})$ be the Heisenberg group, let $\mathbb{H}(\mathbb{Z})$ be its integer subgroup, and let $\mu_H$ be the invariant probability measure on $\mathbb{H}(\mathbb{Z})\backslash\mathbb{H}(\mathbb{R})$. For $f\in C^k(\mathbb{H}(\mathbb{Z})\backslash\mathbb{H}(\mathbb{R}))$, define $\|f\|_{C^k}$ using monomials in the standard left-invariant differential operators of degree at most $k$. Heisenberg-group equidistribution conjecture. There exist $k,l\in\mathbb{N}$ and $\kappa_0\geq2$ such that, for any fixed Diophantine $\alpha$ of type $\kappa\leq\kappa_0$, there is $s=s(\kappa)>0$ for which, for every such $f$ and every $\nu\in C^\infty(\mathbb{R})$ supported on $[-L,L]$, $L\geq1$, the displayed reduced-residue average in the source equals $\widehat{\nu}(0)\int f\,d\mu_H$ with error $O(L\|\nu\|_{C^l}\|f\|_{C^k}c^{-s})$ as $c\to\infty$, where $(a,b)$ solves $ad-bc=1$. The conjecture is a pointwise equidistribution statement intended to imply convergence of the pair-correlation measure to Lebesgue measure; no resolution is supplied in the text.