The symmetric Strassen additivity conjecture for Waring rank

Let $d>1$. Let $F\in\mathbb{C}[x_0,\ldots,x_n]$ and $G\in\mathbb{C}[y_0,\ldots,y_m]$ be non-zero homogeneous forms of degree $d$ in disjoint sets of variables. For a degree-$d$ form $H$, its Waring rank is

$$\operatorname{rk}(H)=\min\{r:H=L_1^d+\cdots+L_r^d\text{ for linear forms }L_i\}.$$

Symmetric Strassen additivity conjecture.

$$\operatorname{rk}(F+G)=\operatorname{rk}(F)+\operatorname{rk}(G).$$

This is the symmetric analogue of Strassen's additive conjecture for tensor rank. The paper proves the equality when either form is a power, when both forms have two variables, or when either form has sufficiently small rank; the general statement remains open.