Buratti-Horak-Rosa conjecture on edge lengths of Hamiltonian paths

Let $[n]=\{1,2,\ldots,n\}$ denote the vertices of a circulant instance, and let edge length $i$ refer to the corresponding circulant edge-length class, for $i=1,\ldots,d$. Let $t_1,t_2,\ldots,t_d$ be nonnegative integers, where $t_i$ counts the edges of length $i$.

Buratti-Horak-Rosa conjecture. If

$$\sum_{i=1}^d t_i=n-1$$

and, for every divisor $q$ of $n$,

$$\sum_{1\leq i\leq d:\,q\mid i}t_i\leq n-q,$$

then there exists a Hamiltonian path on $[n]$ using exactly $t_i$ edges of length $i$ for each $i=1,\ldots,d$.

The conjecture characterizes the combinations of edge lengths that can occur in a Hamiltonian path and asserts that the divisibility inequalities are the only obstruction. Its prime-order case was conjectured by Buratti in the 2000s; the supplied text gives no evidence that the full conjecture has been resolved.