A bandwidth refinement of the Erdős distinct subset sums bound

Let $f(n)$ be the least possible largest element of an $n$-element set of positive integers with pairwise distinct subset sums. Dubroff, Fox and Xu proved the finite lower bound \[ f(n)\ge \binom{n}{\lfloor n/2\rfloor} \] by applying a vertex-boundary estimate to the $2^{n-1}$ subsets whose sums l