IMC 2023 Problem 8

Let $T$ be a tree with $n$ vertices; that is, a connected simple graph on $n$ vertices that contains no cycle. For every pair $u$ , $v$ of vertices, let $d(u,v)$ denote the distance between $u$ and $v$ , that is, the number of edges in the shortest path in $T$ that connects $u$ with $v$ .

Consider the sums $$W(T)=\sum_{\substack{\{u,v\}\subseteq V(T) u\neq v}}d(u,v) \text{and} H(T)=\sum_{\substack{\{u,v\}\subseteq V(T) u\neq v}}\frac{1}{d(u,v)}$$ Prove that $$W(T)\cdot H(T)\geq \frac{(n-1)^3(n+2)}{4}.$$