The topological rigidity conjecture for affine K-systems with hyperbolic centralizers

Let $X$ be a compact homogeneous manifold, let $f_0\in\operatorname{Aff}(X)$ be a $K$-system, and suppose that $\mathcal{Z}(f_0)$ contains a hyperbolic affine diffeomorphism. If $f\in\operatorname{Diff}(X)$ is sufficiently $C^1$-close to $f_0$ and $\mathcal{Z}(f)\doteq\mathcal{Z}(f_0)$, then the topological rigidity conjecture asserts that $f$ is $C^0$-conjugate to an affine $K$-system. The conjecture extends the topological conjugacy expected from hyperbolic examples to affine $K$-systems that need not themselves be hyperbolic; the source emphasizes that structural stability and an Anosov element in the perturbed centralizer are not known a priori.