Multiplicity conjecture for maximum distance sets

Let an $m$-distance set be a set of points whose pairwise distances take at most $m$ values. The multiplicity of a distance is the number of unordered pairs of points in the set realizing that distance, and a maximum $m$-distance set has largest possible cardinality. Multiplicity conjecture. A maximum $m$-distance set will not have a distance with multiplicity less than or equal to $2$. The paper motivates this from computed constructions, but provides no proof of the assertion.