Leading-coefficient conjecture for the $q$-chromatic polynomial of trees

Let $G$ be a finite tree, and let $\widetilde{\chi}_G^{\mathbf{1}}(q,x)$ denote the $q$-chromatic polynomial of $G$ with respect to the all-ones weight vector $\mathbf{1}$. Leading-coefficient conjecture. The leading coefficient of $\widetilde{\chi}_G^{\mathbf{1}}(q,x)$ distinguishes trees. The authors state that they checked this conjecture for all trees with at most $17$ vertices, but no general resolution is given here.