Quantitative inverse Goldbach conjecture for dense sets

Let $\delta>0$. Dense-set inverse Goldbach conjecture. If $X$ is sufficiently large in terms of $\delta$ and $A,B\subset[X]$ satisfy

$$|A|,|B|\geqslant X^\delta,$$

then $A+B$ contains a composite number. This would give a stronger quantitative form of the inverse Goldbach assertion discussed in the paper, but no proof is known.