What is the best upper bound on the running time of the graph minor algorithm? 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.
My answer was based on the explanation in paper of Adler et al for disjoint paths problem. That running time is not exactly what is in the Robertson and Seymour paper but it is the improved one by Kawarabayashi and Wollan. It’s been a long time ago and I am not sure if this is restricted to specific graph class. You may have a look at the corresponding reference.

https://arxiv.org/pdf/1310.2378.pdf