Folklore conjecture on better-quasi-ordering finite graphs by minors

Let a graph be better-quasi-ordered (BQO) under the minor relation when it satisfies the better-quasi-order property for that relation. Folklore conjecture. The finite graphs are better-quasi-ordered (BQO) under the minor relation. This is a well-known open problem. The paper proves that it is equivalent to Thomas' conjecture restricted to rayless graphs, and that it implies that the countable rayless graphs are better-quasi-ordered.