Flat degenerate metrics and Riemannian foliations

Bandyopadhyay, Dacorogna, Matveev and Troyanov conjectured that a closed manifold admitting a flat, non-negative definite metric of constant rank $m$ should be finitely covered by a fiber bundle over the $m$-torus. We give a counter-example to this statement and we discuss the link between this problem and the study of transversely flat Riemannian foliations.