The manifold $g''$-conjecture for balanced homology manifolds

Let $\Delta$ be a balanced, orientable $\mathbf{k}$-homology manifold without boundary. Define $g”_j(\Delta)=h”_j(\Delta)-h”_{j-1}(\Delta)$. Balanced manifold $g”$-conjecture. The vector

$$(g”_j(\Delta))_{j=0}^{\lfloor d/2\rfloor}$$

is the $f$-vector of a simplicial complex. The conjecture is known for barycentric subdivisions of homology manifolds, but remains open in general.