Terai's conjecture on h-vectors of Buchsbaum complexes

A vector $h=(1,h_1,\dots,h_d) \in \mathbb{Z}^{d+1}$ is the $h$-vector of a $(d-1)$-dimensional Buchsbaum complex $\Delta$ such that $\beta_k(\Delta)=0$ for $k \leq d-3$. A vector $(1,h_1,\dots,h_d) \in \mathbb{Z}^{d+1}$ is an $M$-vector if $h_1 \geq 0$ and $0 \leq h_{i+1} \leq h_i^{\langle i \rangle}$ for $i=1,2,\dots,d-1$. Terai's conjecture. The vector $h$ has the stated property if and only if: (a) $(1,h_1,\dots,h_{d-1})$ is an $M$-vector; and (b) $-\frac{1}{d}h_{d-1} \leq h_d \leq h_{d-1}^{\langle d-1\rangle}$. This conjecture gives numerical conditions characterizing the $h$-vectors of Buchsbaum complexes with the specified vanishing Betti numbers; the paper proves the relevant two-dimensional cases, while the general statement is the conjectural assertion.