In dimension $d = 4, 5, 6, 7$, we find the first examples of complete finite volume hyperbolic $d$-dimensional manifolds $M$ with cusps such that an infinite number of orbifolds $M_m$ obtained by generalized Dehn fillings on $M$ admit a properly convex real projective structure. The manifolds $M$ are covering of hyperbolic Coxeter orbifolds and the orbifold fundamental groups $\Gamma_m$ of $M_m$ are Gromov hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d−2}.