Bukh's corrected conjecture on sums of linear transformations

Let $\mathcal{L}_0,\ldots,\mathcal{L}_k\in\operatorname{Mat}_d(\mathbb{Z})$ be irreducible and coprime matrices, and let $A$ be a finite subset of $\mathbb{Z}^d$. Bukh's corrected conjecture. One should have

$$|\mathcal{L}_0 A+\cdots+\mathcal{L}_k A|\geq\left(|\det(\mathcal{L}_0)|^{1/d}+\cdots+|\det(\mathcal{L}_k)|^{1/d}\right)^d|A|-o(|A|).$$

This is a discrete Brunn–Minkowski-type conjecture for sums of linear transformations. The source describes it as a corrected version of Bukh's original conjecture, first stated in work cited as CL23, but does not provide evidence that it has been resolved.