Necklace-bijection explanation for modular subset sums

Determine whether the translation-orbit map from order-fkf subsets of fZ/nZf whose element sum is fkf modulo fnf (when fkf is odd) or fk/2f modulo fnf (when fkf is even) to necklaces with fkf black and fn-kf white beads is bijective for general fnf and fkf, thereby explaining the equality of the two enumeration formulas.

Background

Section 5.1 proves that the number of necklaces with fkf black beads and fn-kf white beads equals fT(n,k,0)f for odd fkf and fT(n,k,k/2)f for even fkf. The authors consider the natural map fγf sending an order-fkf subset of fZ/nZf to its translation orbit. When fnf and fkf are coprime, this map gives a bijection for each fixed sum class. However, the authors report that for many examples with fgcd(n,k)≠1f, the map is not bijective, so a general bijective explanation of the coincident formulas remains unresolved.