On sum-intersecting families of positive integers

We study the following natural arithmetic question regarding intersecting families: how large can a family of subsets of integers from $\{1, \ldots n\}$ be such that, for every pair of subsets in the family, the intersection contains a sum $x + y = z$? We conjecture that any such sum-intersecting family must have size at most $\frac14 \cdot 2^{n}$ (which would be tight if correct). Towards this conjecture, we show that every sum-intersecting family has at most $0.32 \cdot 2^n$ subsets.