Eisenbud–Green–Harris conjecture on quadratic Artinian algebras

Let $K$ be a field, and consider the $h$-vectors of quadratic Artinian $K$-algebras and the $f$-vectors of simplicial complexes. Eisenbud–Green–Harris conjecture. Every $h$-vector of a quadratic Artinian $K$-algebra should be an $f$-vector of a simplicial complex:

$$\{\text{$h$-vectors of quadratic Artinian $K$-algebras}\}\subseteq\{\text{$f$-vectors of simplicial complexes}\}.$$

The conjecture generalizes the preceding flag-complex assertion, which the paper identifies as a particular case. The supplied text gives no resolution of the general conjecture.