Roots-of-unity conjecture for the three-variable power-sum system

Assume $n=3$, $\gcd(a_1,a_2,a_3)=1$, and let $\mathcal{A}_p=\{a_1\bmod p,a_2\bmod p,a_3\bmod p\}$ for $p=2,3$. Let $\zeta$ be a primitive cube root of unity, and let $(\mathrm{SI})$ denote the projective solution set of the system of three power-sum equations at infinity. Roots-of-unity conjecture.

$$(\mathrm{SI})\subseteq\{(1:-1:0),(1:0:-1),(0:1:-1),(1:\zeta:\zeta^2),(1:\zeta^2:\zeta)\}.$$

This strengthens the CKW conjecture: when $0\in\mathcal{A}_2\cap\mathcal{A}_3$, none of these five points occurs, so $(\mathrm{SI})$ is empty. It is verified in the source for $a_1<a_2<a_3$ with $a_1+a_2+a_3\leq300$.