Flaminio–Katok conjecture on smooth stable and unstable foliations

Let $f:M\to M$ be a $C^k$-smooth Anosov diffeomorphism. Denote its stable and unstable foliations by $\mathcal{F}^s_f$ and $\mathcal{F}^u_f$, respectively. Flaminio–Katok conjecture. If both foliations $\mathcal{F}^s_f$ and $\mathcal{F}^u_f$ are $C^2$-smooth, then $f$ is $C^{\max\{2,k\}}$ conjugate to a hyperbolic automorphism of an infra-nilmanifold. The conjecture concerns the rigidity of Anosov diffeomorphisms with highly regular invariant foliations and can be divided into two parts, both of which remain open.