Tuschmann's finite-action conjecture for almost non-negatively curved manifolds

Let $M$ be a manifold with almost non-negative sectional curvature, abbreviated ANSC. The fundamental group $\pi_1(M)$ acts on the homotopy group $\pi_2(M)$ by deck transformations. Tuschmann's finite-action conjecture. If $M$ is ANSC, then there exists a finite-index subgroup of $\pi_1(M)$ that acts trivially on $\pi_2(M)$. This is one of Tuschmann's conjectures concerning the constraints imposed by almost non-negative curvature; the paper discusses rational-model evidence but does not establish the claim.