The $K(\pi,1)$-property for smooth marked curves over finite fields

In the case of smooth marked curves $(X,T)$ over finite fields of characteristic $p$, we study the $K(\pi,1)$-property for $p$. We prove that $(X,T)$ has the $K(\pi,1)$-property if $X$ is affine, and give positive and negative examples in the case when $X$ is proper. We also consider the case of unmarked proper curves over a finite field of characteristic different from $p$.