Ottaviani's conjecture on the generic k-rank of forms

Let $S$ be the polynomial ring in $n$ variables over the complex numbers. For positive integers $k,d,n$, let $\operatorname{rk}_k^\circ(kd,n)$ denote the $k$-rank of a general form of degree $kd$ in $n$ variables. Ottaviani's conjecture.

$$\operatorname{rk}_k^\circ(kd,n)=\begin{cases} \min \left\{s\ge 1\mid s\binom{n+d-1}{n-1}-\binom{s}{2}\ge \binom{n+2d-1}{n-1}\right\}, & k=2,\\ \min \left\{s\ge 1\mid s\binom{n+d-1}{n-1}\ge \binom{n+kd-1}{n-1}\right\}, & k\ge 3. \end{cases}$$

This conjecture predicts the generic length of decompositions into powers of forms and agrees with the known binary-form results and several computed cases; it remains open in general.