Say that a graph is "$d$-dimensional" if it is the node-disjoint union of $1$-skeletons of closed convex polytopes in $d$ dimensions, or a subgraph thereof. So the $2$-dimensional graphs are exactly those that are the disjoint union of cycles and paths, the $3$-dimensional graphs are exactly the planar graphs, and so on. (Is there a standard name for this?)
Clearly for any fixed $d$ the $d$-dimensional graphs are minor-closed, so by the graph minor theorem they can be characterized by finitely many forbidden minors. Is it known what the forbidden minor sets are for general $d$, or even for any $d > 3$? Alternately, is it known that the forbidden minor sets need to be unreasonably large (like how the torus needs $>1000$ forbidden minors)?