Counterexamples to a Conjecture on Laplacian Ratios of Trees

For a graph $G$ with no isolated vertices, its Laplacian ratio is defined as $$\pi (G)=\frac{\mathrm{per}(L(G))}{\prod _{v\in V(G)}d(v)},$$ where $L(G)$ is the Laplacian matrix of $G$, $d(v)$ is the degree of $v$, and $\mathrm{per}$ 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.