The corrected Ruelle zeta-function rationality conjecture

Let $q$ be a prime power, let $T_x^{(n)}$ be the Hecke operators associated with points $x\in\mathbb F_q\setminus\{0,1,t\}$, and let

$$D=(q+1-T_{\tan})^{-1}.$$

Write $D^{(n)}$ for the corresponding operator at the $n$th iteration, and define

$$\operatorname{Corr}(n,k)=-\frac{(-1-q^n)^k}{(1-q^{-n})(1-q^{2n})}.$$

Corrected Ruelle zeta-function conjecture. For any $x_1,\dots,x_k\in\mathbb F_q\setminus\{0,1,t\}$ with $k\ge 1$, the series

$$\exp\left(-\sum_{n\ge 1}\frac{t^n}{n}\left\{\operatorname{Trace}\left(T_{x_1}^{(n)}\cdots T_{x_k}^{(n)}D^{(n)}\right)+\operatorname{Corr}(n,k)\right\}\right)$$

is a rational function.

The correction term is intended to account for missing fixed points in the proposed dynamical interpretation. The source reports this as a conjecture based on computer experiments; without the correction, the analogous series is generally only meromorphic rather than rational.