The minimal excluded minors conjecture for graphs with $g(H)\leq 2$

Let $g(H)$ denote the least integer $k$ such that every $k$-connected $H$-minor-free graph has bounded pathwidth. Let $\mathcal{H}_2$ be the class of graphs $H$ with $g(H)\leq 2$, and let $Q$ be the octahedron graph $K_{2,2,2}$ minus the edges of a triangle. Minimal excluded minors conjecture. The minimal excluded minors for $\mathcal{H}_2$ are

$$\{K_4, K_{2,3}, K_3\cup K_3, Q\}.$$

The conjecture is equivalent to saying that $g(H)\leq 2$ if and only if $H$ is outerplanar and has a vertex transversal of size at most $1$; its resolution is not given in the source.