The flag lower bound conjecture for g-numbers

Let $\Delta$ be a flag homology $(d-1)$-sphere, and let $g_i(\Delta)$ denote its $g$-numbers. Flag lower bound conjecture.

$$g_i(\Delta)\geq \binom{d}{i}-\binom{d}{i-1}.$$

Equality holds if and only if $\Delta$ is the boundary complex of the $d$-cross-polytope. This would give sharp lower bounds for the $g$-numbers of flag homology spheres; the source explicitly states that the conjecture is open.