The site's wording counts ordered pairs, and under that count it fails at every $n\ge2$: each unordered pair at distance $d$ gives two ordered pairs, so every multiplicity is even and the profile $\{f(d_i)\}=\{n-1,\ldots,1\}$, which contains $1$, never occurs. The $n$ equally spaced points $0,1,\ldots,n-1$ on a line, which the question names, then have the multiplicities $2(n-1),\ldots,2$, so the "if" direction fails and the answer is no for a reason that has nothing to do with the problem; the smallest instance is $n=2$, two points with the one multiplicity $2$. The change replaces "ordered pairs" by "unordered pairs"; nothing else changes. The evidence is the counting convention of the sources. Erdős's own statement [Er84c, p. 135] conjectures that the multiplicities $u_1,\ldots,u_m$ of the distinct distances among $n$ points cannot be "a permutation of $1,2,\dots,n-1$" unless the points are equidistant on a line or a circle; such multiplicities sum to $\binom n2$, the number of unordered pairs, and equally spaced points on a line achieve the profile only under that count. [CDL25, Section 5, p. 9] states the question for multiplicities whose sum is $\binom n2$. The site's own commentary credits the arc-and-center family of [CDL25] as a further configuration with the profile, which it is only under the unordered count. The defect is the site's: Erdős's text speaks of the multiplicities of the distances and not of ordered pairs. The form follows from these sources, not from the results that settle it. No result concerns the ordered count alone. The page's standing judges the corrected Statement.