Cubications versus immersions' conjecture

Let $M^n$ be a closed $n$-manifold. A marked cubical decomposition of $M^n$ is a cubical decomposition together with the marking data used to distinguish its cubical cells, and two such decompositions are equivalent modulo cubical flips. Let $N(M^n)$ denote the cobordism group of codimension-one immersions into $M^n$: its elements are cobordism classes of immersions $f:F\to M^n$ from closed $(n-1)$-manifolds, with disjoint union as the group operation. Cubications versus immersions' conjecture. The set of marked cubical decompositions of the closed manifold $M^n$ modulo cubical flips is in bijection with the elements of the cobordism group of codimension-one immersions into $M^n$. The conjecture seeks a complete algebraic-topological classification of cubical decompositions under flip equivalence; the source states that half of it was proved, while the conjecture's solution would give a satisfactory answer to Habegger's problem.