Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6)

Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each $a_n$ lies between $n^2$ and $(n+ 1)^2$. Will there always be a positive integer that can be written in a thousand different ways as a sum of two numbers from the sequence?

I am curious to know current status and references for this problem.