Polynomial-time solvability of Dominating Set at inverse-constant linear radius

Establish whether, for every positive integer k, Sidestep(Dominating Set, dist, n/k) belongs to FP, where dist is the maximum-degree distance between the input graph and the edited graph.

Background

For Dominating Set under the maximum-degree distance, the paper proves NP-hardness at radius n^{1-o(1)} and polynomial-time solvability at radius n/e. The question asks whether tractability extends to every radius of the form n/k for a fixed positive integer k. The case k=e is established in the paper, but the general statement is left unresolved.