The $R_{\infty}$ property for Houghton's groups

We study twisted conjugacy classes of a family of groups which are called Houghton's groups $\mathcal{H}_n$ ($n\in\mathbb{N}$), the group of translations of $n$ rays of discrete points at infinity. We prove that the Houghton's groups $\mathcal{H}_n$ have the $R_\infty$ property for all $n\in \mathbb{N}$.