[2] Later, Huyk Kim and Hyunkoo Lee proved for affine manifolds, and more generally projective manifolds developing into an affine space with amenable holonomy by a different technique using nonstandard polyhedral Gauss–Bonnet theorem developed by Ethan Bloch and Kim and Lee.
[6] In 2008, after Smillie's simple examples of closed manifolds with flat tangent bundles (these would have affine connections with zero curvature, but possibly nonzero torsion), Bucher and Gelander obtained further results in this direction.
In 2015, Mihail Cocos proposed a possible way to solve the conjecture and proved that the Euler characteristic of a closed even-dimensional affine manifold vanishes.
After the correction, their current result is a formula that counts the Euler number of a flat vector bundle in terms of vertices of transversal open coverings.
There also exists a related conjecture by Mikhail Leonidovich Gromov on the vanishing of bounded cohomology of affine manifolds.