A cornerstone of the graph minor theory is an algorithm that, given undirected graphs $G, H$, runs in time $f(|H|)poly(|G|)$, and determines whether $H$ is a minor of $G$ or not. It has been obtained from the famous result of Roberson and Seymour.
This result, as well as other algorithms for minor testing, is always cited with $f$ being some implicit computable function. I wonder what are the best known upper bounds on $f$, e.g., is it elementary?
In this discussion, user @Saeed mentions that $f$ is at most quadruple-exponential but provides no reference for this claim.