Novikov's common-level quasiperiodicity conjecture

Let $M$ be a compact manifold and let several closed $1$-forms on $M$ be given. Their common level is the intersection of the corresponding level sets at a generic choice of levels; a manifold is quasiperiodic when it is assembled from finitely many pieces according to a quasiperiodic pattern. Novikov's conjecture. The generic common level of any number of closed $1$-forms is a quasiperiodic manifold. The source states that, unlike the one-form case, no progress had been made for common levels of two or more closed $1$-forms, so the problem remains open there.