Iarrobino–Kanev conjecture on the small tangent space condition for apolar algebras

Let $S=k[\alpha_1,\ldots,\alpha_n]$ and $P=k[x_1,\ldots,x_n]$ be polynomial rings over a field $k$, with $S$ acting on $P$ by contraction. For a homogeneous polynomial $f\in P$, write $\operatorname{Apolar}(f)=S/\operatorname{Ann}(f)$ for its apolar algebra, and say that it satisfies the small tangent space condition when $S/\operatorname{Ann}(f)^2$ has the smallest possible Hilbert function. Let $d$ be an odd integer. Iarrobino–Kanev conjecture. If one of the following conditions holds

$$\begin{array}{ll} n=4\text{ and }d\geq 15, &\\ n=5\text{ and }d\geq 5, &\\ n\geq 6\text{ and }d\geq 3\text{, except for }(n,d)=(7,3), \end{array}$$

then, for a general homogeneous polynomial $f\in P$ of degree $d$, the apolar algebra $\operatorname{Apolar}(f)$ satisfies the small tangent space condition. The paper proves this conjecture for $d=3$: it holds in all characteristics when $n\geq 18$ and in characteristics $0$, $2$, and $3$ for general $n$; the remaining cases and the full statement for higher odd $d$ are not established here.