OPEN - $500

For what functions $g(N)\to \infty$ is it true that
\[\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}g(N)\]
implies $\limsup 1_A\ast 1_A(n)=\infty$?

It is possible that this is true even with $g(N)=O(1)$, from which the Erdős-Turán conjecture [28] would follow.