Waring-rank inequality for adjoining a variable

Let $f,g\in\mathbb{C}[x_1,\ldots,x_n]$ be homogeneous polynomials of degrees $d$ and $d-1$, respectively. Let $u$ be a new variable, and let $\ell[x_1,\ldots,x_n]$ be the set of all linear forms in $x_1,\ldots,x_n$. Write $\operatorname{WR}(h)$ for the least number of powers of linear forms whose sum is $h$. Waring-rank conjecture. One has

$$\operatorname{WR}(f+ug)\geqslant d+\min_{v\in\ell[x_1,\ldots,x_n]}\operatorname{WR}(f+vg),$$

and equality holds if $g$ is a power of a linear form. This is presented as a potential Waring-rank analogue of the symmetric adjoining conjecture; no proof or resolution is supplied in the given text.