The maximal k-rank conjecture for binary forms

Let $\operatorname{rk}^{\max}_k(kd,2)$ be the least number of $k$-th powers of degree-$d$ forms needed to represent every binary form of degree $kd$. Let $\ell_1$ and $\ell_2$ be non-proportional linear binary forms. Binary maximal k-rank conjecture. For every pair of positive integers $k,d$,

$$\operatorname{rk}^{\max}_k(kd,2)=k,$$

and binary forms of the form $\ell_1\ell_2^{kd-1}$ attain this maximal $k$-rank. This extends the known maximal Waring-rank result for binary forms, but the assertion for arbitrary $k$ and $d$ is presented as conjectural.