The asymptotic broom and $G_{n,k,1}$ conjecture for minimum Wiener-entropy

For integers $n,k,j$, let $G_{n,k,j}$ be the graph formed from a path $P_k$ and a clique $K_{n-k}$ by joining one end vertex of the path to $j$ vertices of the clique. Let a broom be the tree obtained by attaching leaves to one end of a path, and let $I_w$ denote Wiener-entropy. The asymptotic broom and $G_{n,k,1}$ conjecture. There exists a value $n_0$ such that, for every $n\geq n_0$, among all trees of order $n$ the Wiener-entropy is minimized by a broom, while among all graphs of order $n$ it is minimized by a graph of the form $G_{n,k,1}$. The source offers this as a conjecture based on observed extremal behavior for large orders.