Infinite-antichain conjecture for weakly fractal matroid classes

Let $\mathcal M$ be a class of matroids. An antichain is a collection of matroids in which no member is a minor of another. The class $\mathcal M$ is weakly fractal when the sequence measuring the proportion of excluded minors among the class and its excluded minors has an infinite subsequence converging to one. Infinite-antichain conjecture. Any weakly fractal class of matroids contains an infinite antichain. The source observes that all known fractal classes contain infinite antichains and presents this as a proposed general pattern. No resolution is given in the supplied text.