The discretized Erdős ring conjecture

Let $\mathbb{A}$ be an annulus in $\mathbb{R}$, and let $0<\delta\ll1$. Let $A\subset\mathbb{A}$ be a $(\delta,1/2)_1$ set with measure $\approx\delta^{1/2}$. Discretized Erdős ring conjecture. There exists an absolute constant $c_4>0$ such that at least one of $A+A$ and $AA$ has measure

$$\gtrapprox\delta^{\frac12-c_4}.$$

This is a discretized formulation of the question whether a Borel subring of $\mathbb{R}$ can have Hausdorff dimension exactly $1/2$; the source notes that a positive answer to the distance-set improvement conjecture would essentially imply the continuous ring statement.