Cubulated hyperbolic groups admit Anosov representations

We prove that any hyperbolic group acting properly discontinuously and cocompactly on a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ cube complex admits a projective Anosov representation into $\mathrm{S}\mathrm{L}(d,\mathbb{R})$ for some $d$. More specifically, we show that if $\mathrm{\Gamma }$ is a hyperbolic quasiconvex subgroup of a right-angled Coxeter group $C$, then a generic representation of $C$ by reflections restricts to a projective Anosov representation of $\mathrm{\Gamma }$.