Polynomial Stillman bound conjecture for projective dimension

Let $R$ be a polynomial ring over a field, and let $I\subseteq R$ be an ideal generated by $n$ forms of degree at most $d$. Fix $d$, and let $C_d$ be a positive constant depending only on $d$.

Polynomial Stillman-bound conjecture. For all characteristics, the projective dimension of $R/I$ is at worst

$$C_dn^d.$$

This is presented as a conjectural polynomial Stillman bound. The paper notes that the analogous conjecture for ${}^\eta\!A$ is verified for $d\leq 3$, while the Stillman-bound conjecture is not known even for quadrics.