Erdős Problem #41 — Growth of sequences with all triple sums distinct

Here is an old conjecture of mine: Let $a_1 < a_2 < \ldots$ be an infinite sequence for which all the triple sums $a_i + a_j + a_k$ are distinct. Is it then true that $\limsup a_n/n^3 = \infty$? I offer 500 dollars for a proof or disproof of this.