The quadratic Cohen–Macaulay Hirsch conjecture

Let $S=\Bbbk[x_1,\ldots,x_n]$ be a polynomial ring over a field $\Bbbk$, and let $I\subset S$ be a homogeneous ideal generated in degree $2$. An ideal $I$ is Hirsch when

$$\operatorname{diam}(I)\leq \operatorname{height}(I).$$

The quadratic Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

This conjecture generalizes positive results for monomial quadratic ideals satisfying Serre's condition $(S_2)$. Its resolution is not given in the supplied text, so it remains open here.