Random linear Euclidean distance matrix conjecture

Let $M$ be a random linear Euclidean distance matrix of dimension $n$, so that $M$ is an $n\times n$ matrix of pairwise squared distances generated by randomly chosen points on a line. Write $\operatorname{rank}(M)$ for its ordinary rank and $\operatorname{rank}_+(M)$ for its nonnegative rank. Random linear EDM conjecture. With probability one, $\operatorname{rank}(M)=3$ and $\operatorname{rank}_+(M)=n$. The conjecture is motivated by computational experiments and would establish maximal nonnegative rank for generic linear Euclidean distance matrices; the source gives no proof or resolution.