Horak–Rosa conjecture on realizable and admissible chord multisets

Consider $n$ equally spaced points on a circle and write $m=\lfloor n/2\rfloor$. A multiset is represented as $[\ell_1,\ldots,\ell_m]$, where $\ell_j$ counts chords of type $j$. Let $\mathcal{M}_n$ be the class of multisets satisfying $\sum_{j=1}^m\ell_j=n-1$. Let $\mathcal{A}_n\subseteq\mathcal{M}_n$ be the class satisfying, for every divisor $d$ of $n$,

$$\sum_{j=1}^{\lfloor m/d\rfloor}\ell_{jd}\leq n-d,$$

and let $\mathcal{R}_n$ be the class of multisets associated with a path using every point exactly once. Horak–Rosa conjecture. For every $n\geq 1$,

$$\mathcal{R}_n=\mathcal{A}_n.$$

This generalizes Buratti's prime case and characterizes exactly which chord-type multisets are realizable by the necessary admissibility inequalities. The conjecture remains open.