Inverse binary Goldbach conjecture for dense subsets

Let $[N]=\{1,2,\ldots,N\}$, let $\delta>0$, and let $A_1,A_2\subset[N]$ satisfy $|A_1|,|A_2|\geq N^\delta$.

Inverse binary Goldbach conjecture. For $N$ sufficiently large depending on $\delta$, the sumset $A_1+A_2$ contains a composite number.

This is a quantitative inverse problem for additive representations by primes: large subsets of $[N]$ should not have a sumset consisting entirely of primes. The source states that the conjecture is open for $\delta\leq 1/2$.