Nevo's generalized lower bound conjecture for homology spheres

For integers $d,i\geq 1$, let $\mathcal{HS}(i,d)$ be the family of $(d-1)$-dimensional homology spheres without missing faces of dimension greater than $i$. For $\Delta\in\mathcal{HS}(i,d)$, define $g^{(i)}(\Delta):=g^{(d,i)}(h(\Delta,t))$ as the coefficient vector obtained by expressing the $h$-polynomial $h(\Delta,t)$ in the basis

$$B_{d,i}:=(P_{d,i}(t),tP_{d-2,i}(t),t^2P_{d-4,i}(t),\ldots,t^{\lfloor d/2\rfloor}P_{d-2\lfloor d/2\rfloor,i}(t)),$$

where

$$P_{d,i}(t):=(1+t+\cdots+t^i)^q(1+t+\cdots+t^r),$$

and $q\geq 0$, $1\leq r\leq i$ are the unique integers such that $d=qi+r$. Nevo's conjecture. If $\Delta\in\mathcal{HS}(i,d)$, then

$$g^{(i)}(\Delta)\geq 0$$

componentwise. This conjecture generalizes lower-bound results for face numbers of homology spheres; its status is not determined by the supplied source context.