Kalai's lower-bound conjecture for the second g-number of triangulated manifolds

Let $K$) be a connected triangulated $d$-manifold with $d\geq 3$, and let $g_2(K)$ denote its second $g$-number and $\beta_1(K;\mathbb{Q})$ its first Betti number over $\mathbb{Q}$. Kalai's conjecture.

$$g_2(K)\geq\binom{d+2}{2}\beta_1(K;\mathbb{Q}).$$

This gives a topological lower bound on the face numbers of triangulated manifolds; the source attributes it to Kalai, but provides no resolution evidence.