Balog–Shakan conjecture on sums of a set and its dilate

Let $d,q$ be natural numbers with $q>1$, and let $A\subseteq\mathbb{Z}^d$ be a finite $d$-dimensional set.

Balog–Shakan conjecture. One has

$$|A+q\cdot A|\geq(q+2d-1)|A|-O_{q,d}(1).$$

This conjecture extends the bound proved by Balog and Shakan in dimensions $2$ and $3$; the paper presents it as an assertion for all dimensions.