Bajo et al.'s leading-coefficient conjecture for trees

Let $G$ be a graph, and let $\widetilde{\chi}_G^\mathbbm{1}(q,x)$ denote the polynomial introduced in the source for the linear form $\mathbbm{1}=(1,1,\ldots,1)$. Bajo et al.'s conjecture. The leading coefficient of $\widetilde{\chi}_G^\mathbbm{1}(q,x)$ distinguishes trees. The source attributes this conjecture to Bajo et al.; its resolution is not specified here.