A generalization of Gromov's almost flat manifold theorem

Taking as an starting point a question by John Lott about the vanishing of the $\hat A$-genus for spin almost-non-negatively curved mani- folds, we conjecture that an almost-non-negatively curved manifold is either conformally equivalent to a manifold with positive scalar curvature or it is finitely covered by a Nilmanifold. In the way to prove such a claim, we found a generalization of Gromov's almost flat manifold theorem where $L^\infty$ bounds for the curvature are relaxed to mixed curvature bounds. During the talk, we will give the precise statement of our theorem and a detailed sketch of the proof.
This is a joint work with Burkhard Wilking.