Smooth Markov partition conjecture for expanding toral endomorphisms

Let $A$ be an expanding $n\times n$ integer matrix, and let $f$ be the induced toral endomorphism. A smooth Markov partition conjecture asserts that $f$ admits a smooth, in fact linear, Markov partition if and only if there is some $k\in\mathbb{N}$ such that $A^k$ is diagonalizable with integer eigenvalues.

Smooth Markov partition conjecture. The induced toral endomorphism $f$ admits a smooth, in fact linear, Markov partition if and only if for some $k\in\mathbb{N}$, $A^k$ is diagonalizable with integer eigenvalues.

This conjectures a generalization to arbitrary dimensions of the preceding two-dimensional characterization. The paper leaves this generalization for future work; no resolution is given here.