Stanchescu's difference-set lower-bound conjecture in Euclidean space

Let $d \geq 2$ and let $A \subset \mathbb{R}^d$ be a finite set with affine dimension $\dim(A)=d$. Stanchescu's conjecture. Then

$$|A-A| \geq \left(2d-2+\frac{1}{d-1}\right)|A|-(2d^2-4d+3).$$

This conjecture proposes that Stanchescu's construction from $2d-2$ parallel arithmetic progressions gives the best possible lower bound for the size of the difference set of a full-dimensional finite subset of $\mathbb{R}^d$. The bound is known in dimensions $d=1$, $d=2$, and $d=3$, while the higher-dimensional cases were presented as open in the source.