Complete description for the spanning tree problem with one linearised quadratic term

Given an edge-weighted graph the minimum spanning tree problem (MSTP) asks for a spanning tree of minimal weight. The complete description of the associated polytope is well-known. Recently, Buchheim and Klein suggested studying the MSTP with one quadratic term in the objective function resp. the polytope arising after linearisation of that term, in order to better understand the MSTP with a general quadratic objective function. We prove a conjecture by Buchheim and Klein (2013) concerning the complete description of the associated polytope.