Balanced first Betti number lower bound conjecture

Let $\Delta$ be a balanced connected simplicial complex of dimension $d-1 \geq 3$. If $\Delta$ is a ${\bf k}$-homology manifold that is orientable over ${\bf k}$, then the balanced first Betti number conjecture.

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

This is the balanced analogue of the Novik–Swartz lower bound. The inequality is known for the relevant broader class of normal pseudomanifolds in the unbalanced setting, while the balanced formulation remains open.