The universality conjecture for finite connected graph minor classes

Let $X$ be a finite connected graph. A graph is strongly universal for a class of graphs if every graph in the class appears as a subgraph of it, and weakly universal if every graph in the class appears as a minor of it. An $X$-minor-free graph is a graph with no minor isomorphic to $X$.

Universality conjecture. The following are equivalent:

$$\text{There is a strongly universal }X\text{-minor-free graph};$$ $$\text{There is a weakly universal }X\text{-minor-free graph};$$ $$X\text{ is planar}.$$

This conjecture would characterize exactly the finite connected graphs whose minor-closed classes admit universal graphs in either sense. The surrounding discussion notes several positive and negative results, but the equivalence is not established in general.