The forbidden-minor embedding conjecture for graph metrics

Let $L$ be a finite set of graphs. A graph excludes $L$ as a minor if it contains no member of $L$ as a minor, where minors are obtained by edge contractions, edge deletions, and vertex deletions. Forbidden-minor embedding conjecture. There exists a constant $C_L<\infty$ such that every metric on any graph excluding $L$ as a minor can be embedded into $\ell_1$ with distortion at most $C_L$. This conjecture generalizes the Planar Conjecture and remains open despite substantial work on low-distortion embeddings of graph metrics.