Equivalence of primitive-stable and Bowditch actions of the free group of rank two on Gromov-hyperbolic spaces

We prove that the set of Bowditch representations (introduced by Bowditch in 1998, then generalized by Tan, Wong and Zhang in 2008) and the set of primitive-stable representations (introduced by Minsky in 2013) of the free group of rank two in the isometry group of a Gromov-hyperbolic space are equal. The case of $\mathrm{PSL}(2,\mathbb{C})$-representations has already been proved by Lee and Xu and independently Series. Our proof in this context is independent.