Zhang's weak-resonance obstruction conjecture for hyperbolic diffeomorphisms

Let $A$ be a hyperbolic matrix in $GL(n,\mathbb R)$ and let $B$ be a real logarithm of $A$. Let $f(z)=O(|z|^2)$ be $C^{\infty}$ (respectively, analytic), and suppose it has a non-vanishing weakly resonant monomial $w_1^{m_1}\cdots w_n^{m_n}e_j$ with respect to $B$. The resulting locally hyperbolic diffeomorphism is

$$\varphi(z)=Az+f(z).$$

Zhang's conjecture. The diffeomorphism $\varphi$ is not embedded in a $C^{\infty}$ (respectively, analytic) flow.

The conjecture proposes that weak resonances obstruct embedding a hyperbolic diffeomorphism into a smooth or analytic flow. The source states that the conjecture is false and gives two counterexamples consisting of resonant diffeomorphisms.