Transitivity conjecture for strong unstable foliations near the toral automorphism A

Let $A\colon \mathbb T^3\to\mathbb T^3$ be the hyperbolic automorphism induced by the displayed integral matrix, and let $W^{uu}_f$ denote the one-dimensional strong unstable foliation of an analytic diffeomorphism $f$ in a sufficiently small neighborhood of $A$. A foliation is transitive when it has a dense leaf. Transitivity conjecture. For all analytic diffeomorphisms $f$ in a sufficiently small neighborhood of $A$, the strong unstable foliation $W^{uu}_f$ is transitive, that is, it has a dense leaf. The conjecture is supported by numerical calculations of long finite-length strong unstable manifolds, but no resolution is given here.