Maximum Spread of Vertex Degrees in a Simple Graph

We consider the following problem: let $n>k$ be natural numbers, and let $G$ be a graph on $n$ vertices (undirected, without loops or multiple edges). Denote by $h_k(G)$ the number of unordered pairs of vertices in the graph $G$ whose degrees differ by less than $k$. We aim to determine the smallest possible value $f(n,k)$ of the quantity $h_k(G)$. Interest in this question is motivated by the fact that the bipartite analogue of the problem enabled S. Cichomski and F. Petrov to prove the Burdzy – Pitman conjecture on the spread of independent coherent random variables. The problem has been solved under a number of restrictions on $n$ and $k$. A conjecture about the answer in the general case is also presented.