The flag homology sphere conjecture for balanced h-vectors

Let $\Delta$ be a flag homology sphere. The vectors $g(\Delta)$ and $h(\Delta)$ record the face numbers of simplicial complexes, with the latter specifically recording the face numbers of a balanced simplicial complex. Flag sphere face-vector conjecture. The vector $g(\Delta)$ is the $f$-vector of a simplicial complex, and $h(\Delta)$ is the $f$-vector of a balanced simplicial complex. This would strengthen known results relating flag spheres to Frankl–Füredi–Kalai vectors: the corresponding assertion for the $h$-vector follows from the stated proposition, while the $g$-vector assertion is the additional consequence of the flag-sphere conjecture mentioned in the source.