Contracting Boundary of a Cusped Space

Let $G$ be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of $G$ with the topology of fellow travelling quasi-geodesics is compact then $G$ is a hyperbolic group. Let $\mathcal{H}$ be a finite collection of finitely generated infinite index subgroups of $G$. In this article, we prove that if the cusped space $G^h$, obtained by attaching combinatorial horoballs to each left cosets of $H\in\mathcal {H}$, has compact contracting boundary and a vertical ray in each combinatorial horoball is contracting in the cusped space then $G$ is hyperbolic relative to $\mathcal{H}$.