Finitely generated positive cones in $F_n \times \mathbb{Z}$

We construct, for every even $n \ge 2$, a positive cone on $F_n \times \mathbb{Z}$ that is finitely generated as a semigroup, extending the previously known construction for $n = 2$. Malicet, Mann, Rivas and Triestino proved that $F_n \times \mathbb{Z}$ admits an isolated left-order if and only if $n$ is even. Since every finitely generated positive cone determines an isolated left-order, for $n \ge 2$, the group $F_n \times \mathbb{Z}$ admits a finitely generated positive cone if and only if $n$ is even.