Fel'shtyn's rationality conjecture for Reidemeister zeta functions

Let $\Gamma$ be a virtually polycyclic group and let $\varphi:\Gamma\to\Gamma$ be an endomorphism. When the Reidemeister zeta function is well-defined, it is given by

$$R_\varphi(z)=\exp\left(\sum_{n=1}^{\infty}\frac{R(\varphi^n)}{n}z^n\right),$$

where $R(\varphi^n)$ is the number of $\varphi^n$-conjugacy classes. Fel'shtyn's rationality conjecture. For every endomorphism $\varphi:\Gamma\to\Gamma$ of a virtually polycyclic group $\Gamma$, the Reidemeister zeta function $R_\varphi(z)$ is rational whenever it is well-defined. The conjecture extends rationality results known for several classes of virtually polycyclic groups, including finite, abelian, torsion-free nilpotent, and almost-crystallographic groups; the general case remains open.