Canonical representation conjecture for binary forms with a linear coefficient constraint

Let $p$ be a general binary form of even degree $2s$, and let $\alpha_j,\beta_j\in\mathbb{C}$ for $1\leq j\leq s+1$. Canonical representation conjecture. The form $p$ can be written as

$$p(x,y)=\sum_{j=1}^{s+1}(\alpha_jx+\beta_jy)^{2s},\qquad \sum_{j=1}^{s+1}(\alpha_j+\beta_j)=0.$$

This is an alternative version of the preceding canonical-form construction; the source states that it can be verified up to degree $8$, while the general assertion remains open.