Modified maximum subset-divisor conjecture

Let $A$ be a finite set of $n$ positive integers, let $d_k(A)$ be the number of $k$-subsets $B$ of $A$ satisfying $\sum B\mid\sum A$, and let $d(k,n)$ be the maximum of $d_k(A)$ over all such sets $A$. Modified subset-divisor conjecture. For all but finitely many integer pairs $(k,n)$ with $1<k<n$,

$$d(k,n)=\binom{n-1}{k}.$$

This is the paper's corrected form after excluding the infinite families of counterexamples to Huynh's formulation; the paper states that, apart from those families, the conjecture is true.