Erdős's productset conjecture for sets of positive Banach density

Given $A\subseteq\mathbb{Z}$, define its upper Banach density by

$$\operatorname{BD}(A)=\lim_{n\rightarrow\infty}\sup_{m\in\mathbb{Z}}\frac{|A\cap [m+1,m+n]|}{n}.$$

Erdős's productset conjecture. If $\operatorname{BD}(A)>0$, then there are infinite sets $B,C\subseteq\mathbb{Z}$ such that

$$B+C\subseteq A.$$

This conjecture strengthens a conjecture of Erdős about additive structure in sets of positive upper Banach density. The supplied text does not indicate whether it has been resolved.