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 $\{1,\dots n\}$ 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 $\frac{1}{4}\cdot 2^n$ (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.