Generalized lower bound conjecture for vertex-minimal triangulated manifolds

Let $M$ be an $m$-vertex connected closed triangulated $d$-manifold, and let $\beta_i$ be its Betti numbers over some field. Let ${\cal K}_l^{\ast}(d)$ denote the class used in the source. Generalized lower bound conjecture for vertex-minimal triangulated manifolds. For $1\leq l\leq (d-1)/2$,

$${m+l-d-2\choose l+1}\geq {d+2\choose l+1}\sum_{i=1}^l(-1)^{l-i}\beta_i.$$

Moreover, equality holds for some $l<(d-1)/2$ if and only if $M\in{\cal K}_l^{\ast}(d)$. The source presents this as a proposed generalization of a known result and compares it with Kühnel's conjecture; no resolution is reported.