Is the distance geometry problem in NP?

Given a weighted undirected graph with and a positive integer K, the distance geometry problem (DGP) asks to find an embedding of G such that for each edge we have. Saxe proved in 1979 that the DGP is NP-complete with K = 1 and doubted the applicability of the Turing machine model to the case with K > 1, because the certificates for YES instances might involve real numbers. This chapter is an account of an unfortunately failed attempt to prove that the DGP is in NP for K = 2. We hope that our failure will motivate further work on the question.