The cohomology conjecture for blow-up Whitney forms

Let $M$ be a PL manifold with a given triangulation, and let the blow-up Whitney forms on this triangulation be equipped with the continuity conditions described above. Blow-up Whitney cohomology conjecture. The blow-up Whitney forms form a differential complex whose cohomology matches the simplicial cohomology of $M$. This would show that the blow-up construction preserves the topological cohomology of the triangulated manifold; the source provides the claim but no resolution or supporting theorem.