Nonexistence of robustly transitive diffeomorphisms on the 3-sphere

Let $S^3$ denote the three-dimensional sphere. A diffeomorphism is robustly transitive if its transitivity persists under sufficiently small perturbations in the $C^1$ topology. Sphere conjecture. There is no robustly transitive diffeomorphism on $S^3$. The statement is presented as an open question about which manifolds admit robustly transitive diffeomorphisms, and the source gives no resolution.