Damjanović–Wilkinson–Xu affine rigidity conjecture with centralizer isomorphism

Let $G$ be a Lie group, let $X=G/\Gamma$ be a homogeneous space with $\Gamma<G$ cocompact and discrete, and let $f_0=L_a\circ\Psi$ be an affine diffeomorphism, where $a\in G$ and $\Psi$ is a $G$-automorphism preserving $\Gamma$. For a diffeomorphism $f$, write $\mathcal{Z}^\infty(f)$ for its smooth centralizer. Damjanović–Wilkinson–Xu's affine rigidity conjecture. If $f_0$ satisfies the $K$-property and $\mathcal{Z}^\infty(f_0)$ has no rank $1$ factor, then for every $f\in\mathrm{Diff}^\infty(X)$ sufficiently $C^1$-close to $f_0$, if $\mathcal{Z}^\infty(f)\simeq\mathcal{Z}^\infty(f_0)$, then $f$ is smoothly conjugate to an affine diffeomorphism. This asserts local affine rigidity when the smooth centralizer is isomorphic to that of the reference affine system; the supplied text gives no resolution evidence.