Let $A=\{1\leq a_1<a_2<\cdots\}$ and $B=\{1\leq b_1<b_2<\cdots\}$ be sets of integers with $a_n/b_n\to 1$.

If $A+B$ contains all sufficiently large positive integers then is it true that $\limsup 1_A\ast 1_B(n)=\infty$?