The minor-excluded-cover-graph dim-boundedness conjecture

Let $K_t$ denote the complete graph on $t$ vertices, and let a graph exclude $K_t$ as a minor if it has no graph minor isomorphic to $K_t$. The dimension of a poset $P$, denoted $\dim(P)$, is the least number of linear extensions whose intersection is $P$. A class of posets is dim-bounded if the dimensions of its members are bounded by a function of the indicated graph parameter. The minor-excluded-cover-graph conjecture. For every positive integer $t$, the class of posets with cover graphs excluding $K_t$ as a minor is dim-bounded. This statement is presented as a conjectural generalization of the planar cover-graph and bounded-treewidth results described in the paper. The supplied text gives supporting context but no resolution of the conjecture.