Pugh's conjecture on one-dimensional stable manifolds in Pesin theory

Let $f\colon M\to M$ be a $C^1$-diffeomorphism, and let $Orb(p)$ be an orbit with well-defined non-vanishing Lyapunov exponents. In the two-dimensional case, assume that

$$\dim(E^s)=\dim(E^u)=1.$$

Here $E^s$ and $E^u$ are the stable and unstable Lyapunov subspaces along the orbit. Pugh's conjecture. Pesin's result should hold: $W^s(p)$ is a $C^1$ curve tangent at $p$ to $E^s$. More generally, this should hold on manifolds of any dimension whenever $E^s$ has dimension one. The regularity condition in Pugh's formulation, meaning that the Lyapunov exponent is given by the exponential rate of the product of the norm of the derivative, up to replacing $f$ by a large finite iterate, is automatic on one-dimensional subspaces. This conjecture concerns whether Pesin's invariant-manifold conclusion extends to $C^1$ dynamics without the usual $C^{1+\alpha}$ hypothesis; the source does not indicate that it has been resolved.