Spiro's minimum Wiener index conjecture for signed trees

A signed graph is a graph whose edges are assigned positive or negative signs. For a signed graph $(G,\sigma)$, let $W_\sigma(G)$ denote the sum of the signed distances over all unordered pairs of distinct vertices. Let $P_n$ be the path on $n$ vertices, and let $\alpha$ be an alternating signing of $P_n$, meaning that adjacent edges have opposite signs. A tree is a connected graph with no cycles. Spiro's conjecture. Among all signed trees of order $n$, the alternating path $(P_n,\alpha)$ has the minimum Wiener index.

The conjecture concerns the extremal signed Wiener index among trees with a fixed number of vertices. The paper disproves it by constructing an infinite family of counterexamples, proving that it is false whenever $n\ge 30$.