The maximum number of subset divisors of a given size

If s is a positive integer and A is a set of positive integers, we say that B is an s-divisor of A if ∑b∈Bb∣s∑a∈Aa. We study the maximal number of k-subsets of an n-element set that can be s-divisors. We provide a counterexample to a conjecture of Huynh that for s=1, the answer is (n−1k) with only finitely many exceptions, but prove that adding a necessary condition makes this true. Moreover, we show that under a similar condition, the answer is (n−1k) with only finitely many exceptions for each s.