Analytic properties of representation zeta functions of Heisenberg groups

Let $R$ be a ring which is torsion-free and finitely generated over $\mathbb{Z}$, and let $\mathbf{H}(R)$ be the corresponding Heisenberg group. Write $\alpha(\mathbf{H})$ for the abscissa of convergence of its representation zeta function. Heisenberg analytic-properties conjecture. The representation zeta function $\zeta_{\mathbf{H}(R)}(s)$ has abscissa of convergence

$$\alpha(\mathbf{H})=2.$$

It admits analytic continuation to the whole complex plane, and the continued function has no singularities on $\operatorname{Re}(s)=2$ except for a simple pole at $s=\alpha(\mathbf{H})$. This is proposed as a uniform analytic description for Heisenberg groups over torsion-free finitely generated rings; the general assertion is open.