Topological invariance conjecture for zigzag multiplicities and Chern numbers

A zigzag multiplicity is the multiplicity of a zigzag summand in the bicomplex of differential forms of a compact complex manifold, and a Chern number is a characteristic number obtained from its Chern classes. Let $n\geq 3$. Topological invariance conjecture. Modulo universal relations, a linear combination of zigzag multiplicities and Chern numbers of compact complex manifolds in dimension $n$ is an orientation-preserving homeomorphism invariant if and only if it is an orientation-preserving diffeomorphism invariant, if and only if it is a linear combination of the Betti numbers and the Pontryagin numbers. This asks which combinations of the differential-form bicomplex invariants and characteristic numbers depend only on the underlying oriented topological or smooth manifold. Betti numbers and Pontryagin numbers provide the expected topological and smooth invariants, while examples such as deformations of the Iwasawa manifold show that individual odd-zigzag multiplicities need not be topological invariants; the equivalence claimed here remains open.