Degree-two Cohen–Macaulay Hirsch conjecture

Let $S$ be a polynomial ring and let $I\subset S$ be an ideal generated in degree $2$. Write $G(I)$ for the dual graph of the minimal primes of $I$, and call $I$ Hirsch when

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

Degree-two Cohen–Macaulay Hirsch conjecture. If $S/I$ is Cohen–Macaulay, then $I$ is Hirsch.

The conjecture seeks a Hirsch-type diameter bound beyond squarefree monomial ideals, for which a related result is known. The source also records non-Hirsch squarefree monomial examples with Gorenstein quotient, but those examples are generated in high degree.