Flat degenerate metrics and Riemannian foliations

Bandyopadhyay, Dacorogna, Matveev and Troyanov [Math. Z. 305 (2023), p. 25] conjectured that a closed manifold admitting a flat, non-negative definite metric of constant rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> should be finitely covered by a fiber bundle over the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.