Split-factor reformulation of the Eisenbud–Green–Harris conjecture

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$. Let $I$ be a homogeneous ideal containing a regular sequence $f_1,\dots,f_n$ with degrees

$$\deg(f_i)=a_i.$$

A degree-$a_i$ form is said to split into linear factors when it is a product of $a_i$ linear forms. Split-factor reformulation of the Eisenbud–Green–Harris conjecture. $I$ has the same Hilbert function as an ideal containing a regular sequence $g_1,\dots,g_n$ with $\deg(g_i)=a_i$, where every $g_i$ splits into linear factors. This is stated as an equivalent formulation of the EGH conjecture; the reformulation remains open in general.