Stillman's conjecture on uniform projective dimension

Let $d_1,\dots,d_r$ be positive integers. For homogeneous polynomials $f_1,\dots,f_r$ of degrees $\deg(f_i)=d_i$ in a polynomial ring

$$S=\mathbb{C}[x_1,x_2,\dots,x_n],$$

where $n$ is arbitrary, consider the quotient $S/(f_1,\dots,f_r)$ and its projective dimension. Stillman's conjecture. There exists a positive integer $B(d_1,\dots,d_r)$, depending only on the degrees, such that

$$\operatorname{pd}_S S/(f_1,\dots,f_r)\leq B(d_1,\dots,d_r).$$

The conjecture asserts that projective dimension is bounded independently of the number of variables. It is equivalent to saying that the invariant projective dimension is degreewise bounded; the conjecture has since been proved, so its status is recorded as solved.