Stably $C^1$ or stably fractal conjecture for Anosov systems

Let $r\geq 2$ and consider $C^r$ Anosov diffeomorphisms, with their stable and unstable distributions. A distribution is stably $C^1$ if it remains $C^1$ under every sufficiently $C^1$-small perturbation, and stably fractal if every such perturbation has a distribution with a fractal graph. Stably $C^1$ or stably fractal conjecture for Anosov systems. Among $C^r$, $r\geq 2$, Anosov diffeomorphisms, there exists a $C^1$-open, $C^r$-dense set such that the stable or unstable distribution of each element is either stably $C^1$ or stably fractal. This is the Anosov counterpart of the proposed partially hyperbolic dichotomy, and the source gives no resolution.