Characterization conjecture for unavoidable graphs in poset cover graphs

Let $H$ be a graph. Call $H$ unavoidable if the cover graph of every poset of sufficiently large dimension contains $H$ as a minor. Kelly's construction refers to the family of planar cover graphs of pathwidth $3$ arising from Kelly's examples of posets with unbounded dimension. Characterization conjecture for unavoidable graphs. A graph $H$ is unavoidable if and only if $H$ is a minor of some graph from Kelly's construction. Ladders and $K_4$ are known to be unavoidable, while the source describes a full characterization as open and presents Kelly's construction as a necessary restriction.