Smooth conjugacy conjecture for Anosov diffeomorphisms of the three-torus

Let $f$ and $g$ be topologically conjugate $C^r$, $r>1$, Anosov diffeomorphisms of $\mathbb T^3$ with coinciding periodic data, meaning that the differentials of corresponding return maps at every periodic point are conjugate. Smooth conjugacy conjecture. Then the conjugacy $h$ is at least $C^1$. The question is part of the problem of determining whether periodic data form a complete set of smooth-conjugacy moduli; in dimension two the conjugacy is known to be $C^{r-\varepsilon}$, while the three-dimensional case is stated as unknown here.