Brassil-Reichstein's conjecture on a system of cubic equations

Let $a>b\geq 0$ be integers and let $[Z_1:Z_2:Z_3]\in \mathbb{P}^2(\mathbb{Q})$. Brassil–Reichstein's conjecture. The system

$$Z_1^3+Z_2^3+9Z_3^3=0,$$ $$3^aZ_1+3^bZ_2+Z_3=0$$

has no solution $(a,b,[Z_1:Z_2:Z_3])$. This conjecture is used to establish trace conditions for general field extensions of degrees of the form $3^{k_1}+3^{k_2}+3^{k_3}$; its resolution status is not specified in the source.