Converse characterization for Cohen–Macaulay powers of weighted trees

Let $G_\omega$ be a weighted tree. Proposition gives necessary conditions for $I(G_\omega)^n$ to be Cohen–Macaulay for all $n\geqslant 1$: $G$ has a perfect matching $\\{x_1y_1,\ldots,x_ty_t\\}$ in which every $y_i$ is a leaf, together with the stated inequalities and unequal adjacent edge weights. Converse conjecture. These necessary conditions are also sufficient; equivalently, the converse of Proposition is true. This conjecture would characterize weighted trees whose edge-ideal powers are Cohen–Macaulay for every positive power. It is proposed in the paper without a resolution.