The odd-order limit-law conjecture for metric spaces

Let $r\geq 3$ be an odd integer, and let $M_r(n)$ denote the class of finite metric spaces on $[n]$ with distances in the relevant $r$-element distance set. Let $\mathcal{L}_r$ be the language of $r$ binary relation symbols, with elements of $M_r(n)$ viewed as $\mathcal{L}_r$-structures as in Theorem 01thm. Odd-order limit-law conjecture. The class $M_r=\bigcup_{n\in\mathbb{N}}M_r(n)$ has a labeled first-order limit law, but does not have a labeled first-order $0$-$1$ law. This conjecture predicts that odd $r$ exhibits genuinely different first-order asymptotic behavior from even $r$, for which the paper proves a labeled first-order $0$-$1$ law; the existence of the limit law and the failure of the $0$-$1$ law remain open here.