Local counterexample conjecture to smooth conjugacy from coinciding periodic data

Let $\mathcal U$ be the neighborhood of the hyperbolic toral automorphism considered in the source. Local counterexample conjecture. For any $f\in\mathcal U$ there exists $g\in\mathcal U$ with the same periodic data which is not $C^1$-conjugate to $f$. The source constructs counterexamples near a particular model and explains that the strong stable and unstable foliations need not match, but states that it does not know whether the counterexample extends to the whole neighborhood $\mathcal U$.