Ananyan–Hochster's quadratic bound conjecture for projective dimension

Let $S$ be a polynomial ring over a field, and let $I\subseteq S$ be an ideal generated by $n$ quadrics and of height $h$. Ananyan–Hochster's quadratic bound conjecture. One has

$$\operatorname{pd}_S(S/I)\le h(n-h+1).$$

Ananyan and Hochster's explicit bound for $n$ quadrics is exponential in $n$, whereas this conjecture predicts a quadratic bound. The conjecture is known when $h\le 2$ and is necessarily sharp in general; its status beyond these cases remains open.