Higher-regularity conjecture for conjugacies near irreducible toral automorphisms

Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\geq 3$, with simple real spectrum and irreducible characteristic polynomial over $\mathbb Z$. Let $\mathcal U$ be the neighborhood from Theorem A, and let $f\in\mathcal U$ satisfy Property $\mathcal A$ and $g\in\mathcal U$ have the same periodic data. Higher-regularity conjecture. In the context of Theorem A, one can actually conclude that $f$ and $g$ are $C^{r-\varepsilon}$ conjugate, where $\varepsilon$ is an arbitrarily small positive number. Theorem A gives only $C^{1+\nu}$ conjugacy under its stated assumptions; the conjecture asserts the same near-optimal regularity known in dimension two.