Erdős Problem #1145 — Representations in Asymptotically Equivalent Sets

Let $A,B\subseteq\mathbb N$ be infinite sets, and let $a_n$ and $b_n$ be their increasing enumerations, so that $a_n/b_n\to1$. Suppose that $A+B$ contains every sufficiently large natural number, where $A+B=\{a+b:a\in A,\ b\in B\}$. For $n\in\mathbb N$, let $(1_A*1_B)(n)$ be the number of pairs $(a,b)\in A\times B$ with $a+b=n$. Is

$$\limsup_{n\to\infty}(1_A*1_B)(n)=\infty?$$