Stillman's conjecture on projective dimension of polynomial-ring quotients

Let $K$ be a field and let $R=K[x_1,\ldots,x_N]$ be a polynomial ring. For fixed positive integers $n$ and $d_1,\ldots,d_n$, let $I\subseteq R$ be an ideal generated by $n$ homogeneous polynomials of degrees $d_1,\ldots,d_n$, and let $\operatorname{pd}(R/I)$ denote the projective dimension of $R/I$ as an $R$-module. Stillman's conjecture. There is an upper bound for $\operatorname{pd}(R/I)$ depending only on $n$ and $d_1,\ldots,d_n$, and independent of the number of variables $N$. This conjecture concerns uniform homological bounds for ideals with fixed numbers and degrees of generators; the paper proves the claim for ideals generated by quadratic polynomials, while the general statement was open in the source.