Lefschetz-type conjecture for the base-change sum of L-functions

Assume the setup in which $G$ is simply connected and almost simple, $S$ is a set of places with at least two elements, and $T=\emptyset$. For each positive integer $m$, let $(X_m,S_m)$ denote the base change of $(X,S)$ to $\mathbb{F}_{q^m}$, and let $\mathcal{L}(G,m)$ be the sum of $L$-functions $L_{S_m}(M_{G_\gamma})$ over the relevant semisimple conjugacy classes $[\gamma]$ for the base-changed data. A function $f\colon\mathbb{Z}\to\mathbb{C}$ is of Lefschetz type if

$$f(m)=\sum_{i=1}^r n_i\alpha_i^m$$

for some $n_i\in\mathbb{Z}$ and $\alpha_i\in\mathbb{C}$. The Lefschetz-type base-change conjecture. The function $m\mapsto\mathcal{L}(G,m)$ is of Lefschetz type.