Wu–Dong–Lai conjecture on Laplacian ratios of trees

Let $T$ be a tree on $n$ vertices. For a graph $G$ with no isolated vertices, define its Laplacian ratio by

$$\pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)}d(v)},$$

where $L(G)$ is the Laplacian matrix, $d(v)$ is the degree of $v$, and $\operatorname{per}$ denotes the permanent. Let $S(n,a)$ and $S(n,a,b)$ denote the tree families used in the conjecture. Wu–Dong–Lai conjecture.

Let $T$ be a tree on $n$ vertices.

(i) If $n$ is odd, then

$$\pi(T)\leq 2\left(\frac{3}{2}\right)^{(n-3)/2},$$

with equality if and only if

$$T=S\left(n,\frac{n-1}{2}\right).$$

(ii) If $n=4k$, then

$$\pi(T)\leq \frac{4n^2+8n+24}{n(n+4)}\left(\frac{3}{2}\right)^{(n-6)/2},$$

with equality if and only if

$$T=S\left(n,\frac{n+2}{2},\frac{n-2}{2}\right).$$

(iii) If $n=4k+2$, then

$$\pi(T)\leq \frac{4n^2+8n+40}{(n+2)^2}\left(\frac{3}{2}\right)^{(n-6)/2},$$

with equality if and only if

$$T=S\left(n,\frac{n}{2},\frac{n}{2}\right).$$

The conjecture proposes the extremal tree and maximum Laplacian ratio for each parity class of $n$. The paper's abstract states that the conjecture has infinite families of counterexamples, so it is refuted.