The balanced generalized lower bound conjecture

Let $\Delta$ be a balanced $d$-polytope, or more generally a balanced $\mathbf{k}$-homology $(d-1)$-sphere. Balanced GLBC.

$$1=\frac{h_0(\Delta)}{\binom d0}\leq\frac{h_1(\Delta)}{\binom d1}\leq\frac{h_2(\Delta)}{\binom d2}\leq\cdots\leq\frac{h_{\lfloor d/2\rfloor}(\Delta)}{\binom d{\lfloor d/2\rfloor}}.$$

The inequalities were established for all balanced polytopes, while the homology-sphere case remains open in the supplied text.