We consider a reaction-diffusion system modeling the co-invasion of a gene drive (an allele biasing inheritance, increasing its own transmission to offspring) and a brake (an "antidote" to a gene drive) in a population carrying a wild-type allele. We successfully prove that, whenever the drive fitness is at most 50% of the wild-type one while the brake fitness is close to the wild-type one, co-extinction of the brake and the drive occurs in the long run. On the contrary, if the drive fitness is at least 50% of the wild-type one, then co-extinction is impossible. Based on numerical experiments, we argue in favor of a global co-extinction conjecture provided the drive fitness is at most 50% of the wild-type one. The proof relies upon the study of a related predator-prey system with strong Allee effect on the prey.