An instance of the quadratic minimum spanning tree problem (QMSTP) is called linearizable if it can be rewritten as an instance of the linear minimum spanning tree problem in such a way that the objective function value is preserved at all feasible solutions. Previous work has shown that a sufficient condition for linearizability is that the (symmetric) matrix of cost coefficients is a weak sum matrix. This condition has been shown to also be necessary for QMSTP instances defined on complete graphs and complete bipartite graphs whose two parts each have at least three vertices, but not necessary for QMSTP instances defined on certain other types of graphs. In this paper, we exploit a recent polyhedral framework for studying linearizable quadratic combinatorial optimization programs to prove the following: among simple connected graphs with at least four edges, the graphs for which every linearizable QMSTP instance has a weak sum symmetric cost matrix are exactly the 3-connected graphs.