Nilpotent geodesic-flow asymptotic expansion conjecture

Let $M$ be a compact Riemann manifold with negative curvature. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let $\Phi:\pi_1(M)\to\Gamma$ be surjective, and let $\alpha$ be a conjugacy class of a central element of $\Gamma$. Let $d$ denote the polynomial growth order of $\Gamma$, and let $C$ be the constant in the nilpotent Floquet–Bloch asymptotic conjecture for the geodesic flow on $N=UM$. Nilpotent geodesic-flow expansion conjecture. Then

$$\pi(x,\Phi,\alpha)\sim\frac{Ce^{hx}}{x^{1+d/2}}\left(1+\frac{c_1}{x}+\frac{c_2}{x^2}+\cdots\right),$$

where the constants $c_1,c_2,\ldots$ are expressed in terms of $M$, via Lie integrals or Chen's iterated integrals, and are explicitly computable. The claim predicts a full asymptotic expansion refining the leading nilpotent orbit-counting term; the supplied source gives no resolution.