Nonnegative BH representation conjecture for Eigen-CG inequalities

For any $(v_0,v)\in\mathbb{R}^{n+1}$, let $\text{E-CG}(v_0,v)$ denote the corresponding Eigen-CG inequality, and let BH inequalities denote the Boros–Hammer inequalities. Nonnegative BH representation conjecture. The inequality defined by $\text{E-CG}(v_0,v)$ is implied by a nonnegative combination of BH inequalities. The preceding argument establishes this implication in a special construction, but the general assertion remains unproved.