Klee–Novik's equality characterization conjecture for balanced 3-manifolds

Let $\Delta$ be a connected $3$-dimensional balanced $\mathbb{F}$-homology manifold. The Klee–Novik conjecture. Equality

$$2f_1(\Delta)-9f_0(\Delta)=24\bigl(\widetilde{\beta}_1(\Delta;\mathbb{F})-1\bigr)$$

holds if and only if $\Delta$ belongs to the balanced Walkup class. The balanced Walkup class consists of balanced simplicial complexes obtained from the boundary $\partial\mathcal{C}_4$ of the $4$-dimensional cross-polytope by successive connected sums with $\partial\mathcal{C}_4$ and balanced handle additions. This conjecture completes the equality characterization in the balanced lower bound theorem, which is known in dimensions at least $4$ but remains unresolved for dimension $3$.