The forbidden-minor characterization of strictly metrizable graphs

A graph is strictly metrizable if every pair of vertices has a unique shortest path under some positive edge weighting. The forbidden-minor conjecture. A graph is strictly metrizable if and only if it contains none of the six graphs in the designated figure as a minor. This would give a finite forbidden-minor characterization of strict metrizability, in the spirit of the graph minor theorem; the source presents it as an open question, and no resolution is supplied.