The Planar Conjecture on embedding planar graph metrics into \ell_1

A planar graph metric is a shortest-path metric on a finite planar graph whose edges have arbitrary weights. The Planar Conjecture. Every metric supported on a finite planar graph can be embedded into $\ell_1$ with constant distortion. This is a central open problem in the theory of metric embeddings; the cited work proves the analogous statement for graphs excluding $K_4$ as a minor, with distortion at most $14$, but the planar case remains unresolved.