Equivalence of constructed obstruction families for composed parameters H–p

Prove that for any minor-monotone graph parameter p with a finite universal obstruction family, the constructed obstructing set 3H is equivalent (under the Smyth extension ordering) to the family 5H as a universal obstruction for the composed parameter H–p.

Background

The authors outline a general composition mechanism to build universal obstructions for parameters of the form H–p, where p lies between treewidth and size and has a finite universal obstruction set. They introduce two families, 3H and 5H, arising from this construction.

They explicitly conjecture that the two constructed families are equivalent as obstructing sets, which would unify and simplify the description of universal obstructions for a broad class of composed parameters.