Generalised lower bound conjecture for neighbourly triangulated manifolds

Let $M$ be an $m$-vertex connected triangulated closed $d$-manifold with Betti numbers $\beta_i$, and let ${\cal K}_l^{\ast}(d)$ denote the class of $(l+1)$-neighbourly members of ${\cal K}_l(d)$. Neighbourly generalised lower bound conjecture. For $1\leq l\leq (d-1)/2$, one has

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

Moreover, equality for some $l<(d-1)/2$ holds if and only if $M\in {\cal K}_l^{\ast}(d)$. This is presented as a proposed extension of a lower-bound theorem for members of ${\cal W}_1(d)$ and is compared with a conjecture of Kühnel.