Locality conjecture for small-doubling sets

Let $A\subset\mathbb{R}^n$ be measurable and let $\delta<\Delta$. Locality conjecture. There is an absolute constant $\Delta>0$ such that if

$$\left|\frac{A+A}{2}\right|\leq(1+\delta)|A|,$$

then there exists a convex set $K\subset\mathbb{R}^n$ with

$$|K\triangle A|\leq O_{\delta}(1)|A|.$$

This is a conjecture recalled from van Hintum, Spink and Tiba concerning the structure of sets with small doubling. The source gives no resolution status.