Conjecture on extensions of nonflat sets in CN2 manifolds

Let $M^n$ be a complete CN2 manifold with finite volume, let $V$ be its set of nonflat points, and let an extension of $V$ be an open set carrying a complete flat totally geodesic distribution that restricts to the nullity distribution on $V$. An extension is maximal if it has no larger extension, and dense if it is dense in $M^n$. Extension conjecture. If $V$ admits a dense, not necessarily locally finite, extension, then the complement of any maximal extension is a disjoint union of compact totally geodesic flat hypersurfaces, possibly accumulating. This conjecture concerns the structure of the flat regions remaining after extending the nonflat locus and is presented as an expected description without a resolution in the source.