Terai's conjecture on sequential Cohen–Macaulay edge ideals of weighted graphs

Let $G$ be a Cohen–Macaulay very well-covered graph, and let $S=K[x_1,\ldots,x_n]$ be a standard graded polynomial ring over a field $K$. For a weight function $\mathbf w:E(G)\to\operatorname{\mathbb{Z}}_{>0}$, define the edge ideal

$$I(G_{\mathbf w})=\big((x_ix_j)^{\mathbf w(x_ix_j)}\mid\{x_i,x_j\}\in E(G)\big)\subseteq S.$$

Terai's conjecture. The ideal $I(G_{\mathbf w})$ is sequentially Cohen–Macaulay for every weight function $\mathbf w$. This conjecture extends questions about Cohen–Macaulayness and sequential Cohen–Macaulayness of edge ideals of edge-weighted graphs. The source presents it as a conjecture attributed to Terai; its resolution is not specified in the supplied text.