Balanced first Betti number equality conjecture

Let $\Delta$ be a balanced connected ${\bf k}$-homology manifold that is orientable over ${\bf k}$ of dimension $d-1 \geq 3$. Let $\mathcal{BH}^d$ denote the class of balanced connected sums of cross-polytopal spheres and balanced handles. The balanced first Betti number equality conjecture.

$$\overline{g}_2(\Delta) \geq 4\binom{d}{2} \beta_1(\Delta;{\bf k})$$

and equality holds if and only if $\Delta\in\mathcal{BH}^d$.

Balanced handles attain equality, motivating this strengthening of the lower-bound conjecture. The claim is presented as open; the cited balanced results provide supporting evidence but do not establish the characterization.