Stably $C^1$ or stably fractal conjecture for partially hyperbolic systems

Let $f$ be a $C^r$ partially hyperbolic diffeomorphism on a compact manifold $M$, with $r\geq 2$. The distribution $E^s_f$ is stably $C^1$ if, for every $\tilde f\in\operatorname{Diff}^r(M)$ that is $C^1$-close to $f$, $E^s_{\tilde f}$ remains $C^1$; it is stably fractal if, for every such $\tilde f$, $E^s_{\tilde f}$ has a fractal graph. The analogous properties are defined for $E^u$ and $E^c$. Stably $C^1$ or stably fractal conjecture for partially hyperbolic systems. Among $C^r$, $r\geq 2$, partially hyperbolic diffeomorphisms, there exists a $C^1$-open, $C^r$-dense set $\Omega$ such that for each $f\in\Omega$, the distribution $E^\ast_f$ for $\ast\in\{s,u,c\}$ is either stably $C^1$ or stably fractal. Moreover, if $E^u_f$ and $E^s_f$ are both non-trivial, then $E^c_f$ is stably fractal. The conjecture is motivated by the expected $C^1$-stability of transversality-driven fractality and the fragility of $C^1$ center distributions; the source gives no resolution.