Iarrobino–Kanev conjecture on Hilbert functions of squared point ideals

Let $n\ge3$, $d\ge2$, and define

$$N_j=\binom{n+j-1}{n-1}.$$

Let $Z$ be a set of $s$ general points in $\mathbb{P}^{n-1}$, where $N_{d-1}\le s<N_d$, and let $I=I(Z)$ be its vanishing ideal. Write $h_{2d}(I^2)$ for the degree-$2d$ Hilbert function of $I^2$. Iarrobino–Kanev conjecture. Except for $(n,d,s)=(3,2,5)$, $(4,2,9)$, and $(5,2,14)$, one has

$$h_{2d}(I^2)\ge\max\left\{ns,\,N_{2d}-\binom{N_d-s+1}{2}\right\}.$$

For the three exceptional triples, the maximum is to be replaced by the minimum. This conjecture predicts the relevant Hilbert function of the square of a general finite set of points and is related to dimensions of tangent spaces to catalecticant varieties. The surrounding discussion presents it as the conjectural input needed to extend the sum-of-squares arguments to at least four variables; its resolution status is not specified in the source.