Existence conjecture for Buchsbaum Stanley–Reisner rings with linear resolution

Let $d$, $c$, $q$, and $h$ be integers with $c\geq 1$, $h\geq 0$, and $2\leq q\leq d$. A Buchsbaum Stanley–Reisner ring is a ring $A=k[\Delta]$ associated with a simplicial complex $\Delta$; its dimension, codimension, and local cohomology are denoted by $\dim A$, $\operatorname{codim} A$, and $H_{\mathfrak m}^{q-1}(A)$, respectively. The ring has a $q$-linear resolution when its minimal free resolution is $q$-linear. Define

$$h_{c,d,q}=\frac{(c+q-2)\cdots(c+1)c}{d(d-1)\cdots(d-q+2)}.$$

Existence conjecture. The following conditions are equivalent:

1. There exists a Buchsbaum Stanley–Reisner ring $A=k[\Delta]$ with $q$-linear resolution such that $\dim A=d$, $\operatorname{codim} A=c$, and $\dim H_{\mathfrak m}^{q-1}(A)=h$.
2. The inequality

$$0\leq h\leq h_{c,d,q}$$

holds.

The preceding result proves the upper bound as a necessary condition. The conjecture asserts that this numerical bound is also sufficient for the existence of such a ring, thereby characterizing the possible values of the local-cohomology invariant $h$.