The conjectured optimal error term for sums of linear transformations

Let $\mathcal{L}_1,\ldots,\mathcal{L}_k\in\operatorname{Mat}_d(\mathbb{Z})$ be irreducible and coprime. The optimal error-term conjecture. There is a constant $D$ such that, for every finite subset $A$ of $\mathbb{Z}^d$,

$$|\mathcal{L}_1 A+\cdots+\mathcal{L}_k A|\geq\left(|\det(\mathcal{L}_1)|^{1/d}+\cdots+|\det(\mathcal{L}_k)|^{1/d}\right)^d|A|-D|A|^{1-1/d}.$$

The paper gives examples showing that an error of order $|A|^{1-1/d}$ can occur and conjectures, following Shakan, that substantially worse errors do not occur. The corresponding assertion is presented as open.