Minimum weighted Szeged index graphs are attained by trees

Let $G$ be an $n$-vertex graph, and let $wSz(G)$ denote its weighted Szeged index. Tree attainment conjecture. The minimum weighted Szeged index among $n$-vertex graphs is attained by a tree. This would reduce the problem of characterizing minimum weighted Szeged index graphs to the study of trees; the paper presents computational evidence and structural results for minimum weighted Szeged index trees, but does not resolve the conjecture.