Weil rationality for difference zeta functions

Let $(X,\sigma)$ be a finite-dimensional difference scheme of finite $\sigma$-type over a finite difference field $(k,\varphi_0)$, where $\varphi_0$ is the distinguished Frobenius on $k$. Its zeta function is defined by

$$Z((X,\sigma)/(k,\varphi_0),t)=\exp\left(\sum_{n\geq1}\frac{|X(\bar{k},\varphi_n)|}{n}t^n\right),$$

with $\varphi_n$ the corresponding powers of Frobenius. Weil rationality for difference zeta functions. The zeta function is near-rational, meaning that its logarithmic derivative is rational. This is a difference-algebraic analogue of Weil's rationality theorem for zeta functions; the supplied text gives no evidence establishing whether the conjecture has been resolved.