Erdős Problem #28 — Unbounded representation counts for additive bases of order 2

Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence of integers, and denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Also Turán and I conjectured that if $f(n) > 0$ for all $n > n_0$, then $\limsup f(n) = \infty$ and perhaps even

$$\limsup f(n)/\log n > 0.$$

I offer 500 dollars for a proof or disproof of my conjecture with Turán.