Refuting a conjecture of Goemans on bounded degree spanning trees

In 2006, Goemans presented an approximation algorithm for the minimum bounded degree spanning tree problem that constructs a tree with cost at most the optimal value of an LP relaxation but degree bound violations of up to two units per vertex. He conjectured that violations of at most one per vertex are attainable, providing a second conjecture that would make his approach achieve this guarantee. While the first conjecture was answered positively by Singh and Lau, we refute the second.