Balanced generalized lower bound conjecture for simplicial polytopes

Let $P$ be a balanced simplicial $d$-polytope, meaning that its underlying graph is $d$-colorable. For $-1\leq i\leq d-1$, let $f_i(P)$ be the number of $i$-dimensional faces, with $f_{-1}(P)=1$, and define

$$h_i(P)=\sum_{j=0}^i(-1)^{j-i}\binom{d-j}{i-j}f_{j-1}(P)$$

for $0\leq i\leq d$. Balanced generalized lower bound conjecture. Then

$$\frac{h_0(P)}{\binom{d}{0}}\leq\frac{h_1(P)}{\binom{d}{1}}\leq\cdots\leq\frac{h_{\lfloor d/2\rfloor}(P)}{\binom{d}{\lfloor d/2\rfloor}}.$$

Moreover, for some $i\leq d/2$, equality

$$\frac{h_{i-1}(P)}{\binom{d}{i-1}}=\frac{h_i(P)}{\binom{d}{i}}$$

is equivalent to $P$ having the balanced $(i-1)$-stacked property. The conjecture is the balanced analogue of the generalized lower bound theorem for simplicial polytopes; the source states that it was proposed by Klee and Novik and that the article aims to prove it.