Fröberg–Iarrobino conjecture on Hilbert functions of generic power ideals

Let $S$ be a polynomial ring in $n$ variables over $\mathbb C$, and let $\ell_1,\ldots,\ell_r$ be generic linear forms. For a positive integer $d$, set $I=(\ell_1^d,\ldots,\ell_r^d)$ and $R=S/I$. Let $\mathrm{HF}_R(t)$ denote the Hilbert series prescribed by Fröberg's formula for $r$ generic forms of degree $d$. Fröberg–Iarrobino conjecture. The Hilbert function of $R$ is the Fröberg-predicted one, except for $(n,r)=(3,7),(3,8),(4,9),(5,14)$ and possibly for $r=n+2$ and $r=n+3$. The conjecture is still largely open, with progress for low degrees and numbers of variables and reformulations using fat points and linear systems.