The torus-fibration conjecture for closed manifolds with flat degenerate metrics

Let $M$ be a closed manifold carrying a flat, possibly degenerate, non-negative definite metric $g$ of rank $m$.

Torus-fibration conjecture. $M$ is finitely covered by a manifold diffeomorphic to a fiber bundle over an $m$-dimensional torus.

This conjecture proposes a global structural description for manifolds with flat possibly degenerate metrics, extending the paper's local results on flat coordinates. The supplied source does not indicate that the conjecture has been resolved.