Counterexamples to a Conjecture on Laplacian Ratios of Trees

For a graph $G$ with no isolated vertices, its Laplacian ratio is defined as $$\pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)},$$ where $L(G)$ is the Laplacian matrix of $G$, $d(v)$ is the degree of $v$, and $\operatorname{per}$ denotes the permanent. Brualdi and Goldwasser asked for the maximum value of $\pi(T)$ among trees $T$ with a fixed number of vertices. Wu, Dong and Lai recently proposed a conjectural answer to this problem. We give infinite families of counterexamples to their conjecture.