The $K_{n+5}$ and $K_{3^2,1^n}$ families conjecture for $n$-apex obstructions

A graph is $n$-apex if deleting $n$ or fewer vertices results in a planar graph. A graph is a minor of another graph if it can be obtained by contracting edges in a subgraph. Let $\mathcal{O}_n$ be the set of graphs that are not $n$-apex although every proper minor is. The $K_{n+5}$ and $K_{3^2,1^n}$ families conjecture. For each $n>0$, the $K_{n+5}$ and $K_{3^2,1^n}$ families are in $\mathcal{O}_n$. Here $K_{n+5}$ is the complete graph on $n+5$ vertices, $K_{3^2,1^n}$ is the complete multipartite graph with two parts of three vertices and a further $n$ singleton parts, and each family consists of graphs obtained through zero or more triangle-Y or Y-triangle moves. The conjecture would identify these families as minor-minimal obstructions to the $n$-apex property and provides evidence toward determining the finite obstruction sets $\mathcal{O}_n$; the paper reports computational evidence but does not establish the claim in general.