Conjecture on lines in metric spaces with finitely many distances

Let $(V,\rho)$ be an $n$-point metric space with $n\geq 2$, and suppose that the number of distinct distances occurring in the metric is bounded by an absolute constant. Conjecture on finitely many distances. Such a metric space has $\Omega(n^{4/3})$ lines. The paper proves an $\Omega(n)$ bound for a constant number of distinct distances and establishes stronger results for particular distance sets, so the conjectured exponent remains open.