Díaz–Gorodetski global non-hyperbolic measure conjecture

Let $M$ be a compact smooth manifold without boundary, and let $\operatorname{Diff}^r(M)$ denote the space of $C^r$ diffeomorphisms of $M$, where $r\geq 1$. An invariant measure is non-hyperbolic if it has a vanishing Lyapunov exponent. Díaz–Gorodetski's conjecture. There is an open dense subset $\mathcal{U}\subset\operatorname{Diff}^r(M)$ where $r\geq 1$, such that every diffeomorphism $f\in\mathcal{U}$ either is uniformly hyperbolic or has an ergodic non-hyperbolic invariant measure. The paper states this as a typical-dynamics version of the converse to the implication that a non-hyperbolic ergodic measure rules out uniform hyperbolicity, and later says that it has not been resolved there.