Gogolev–Maimon–Kolgomorov conjecture on transitivity of the strong unstable foliation

Let $A$ be a hyperbolic automorphism of the $3$-torus with three real eigenvalues satisfying

$$|\lambda_1|<1<|\lambda_2|<|\lambda_3|.$$

Gogolev–Maimon–Kolgomorov conjecture. For all analytic diffeomorphisms $f$ in a sufficiently small neighborhood of $A$, the strong unstable foliation ${\mathcal W}^{uu}$ is transitive, meaning that it has a dense leaf. The paper states that this conjecture is answered positively, so the asserted property holds for the indicated neighborhood.