Well-quasi-ordering conjecture for finite-tree-width graphs

Let a graph have finite tree-width if its tree-width is finite, and consider the minor relation on graphs. Finite-tree-width well-quasi-ordering conjecture. The graphs of finite tree-width are well-quasi-ordered under the minor relation. This is presented as an even stronger conjecture motivated by the fact that known counterexamples to well-quasi-ordering infinite graphs do not have finite tree-width. Its status is refuted according to the supplied evidence.