The star conjecture for maximum Wiener-entropy among trees

Let $T$ be a tree of order $n\geq 5$, and let $S_n$ denote the star on $n$ vertices. The Wiener-entropy $I_w(T)$ is defined from the vertex transmissions of $T$. The star conjecture. The inequality

$$I_w(T)\leq I_w(S_n)$$

holds, with equality if and only if $T=S_n$. This conjecture, attributed in the source to Dehmer and Kraus, is disproved in the paper.