Small-product-set difference containment conjecture

Let $X\subset\mathbb{R}$ be a finite real set, and let $XX=\{xy:x,y\in X\}$. Let $A\subset\mathbb{R}$ be a set with difference set $A-A=\{a-a':a,a'\in A\}$. Small-product-set difference containment conjecture. For every $\epsilon>0$ there exists $\delta>0$ such that, for every real set $X$ satisfying

$$|XX|\leq |X|^{1+\delta},$$

if

$$A-A\subset X,$$

then

$$|A|\ll_\delta |X|^\epsilon.$$

This is presented as an alternative form of the iterated difference-product conjecture: a set with very small multiplicative doubling should not contain the difference set of a substantially larger set. The source gives no resolution.