The improved Bukh conjecture on sums of irreducible transformations

Let $\operatorname{Mat}_d(\mathbb{Z})$ denote the integer $d\times d$ matrices, and let irreducible mean that the transformations have no common non-trivial invariant subspace in the sense used in the paper. For $\mathcal{L}\in\operatorname{Mat}_d(\mathbb{Q})$, let $f(x)=\prod_{i=1}^d(a_i x+b_i)$ be its minimal polynomial over $\mathbb{Z}$ with coprime coefficients, and define

$$H(\mathcal{L})=\prod_{i=1}^d(|a_i|+|b_i|).$$

The improved Bukh conjecture. Let $\mathcal{L}_1,\mathcal{L}_2\in\operatorname{Mat}_d(\mathbb{Z})$ be irreducible. Then, for every finite subset $A$ of $\mathbb{Z}^d$,

$$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq H(\mathcal{L}_1^{-1}\mathcal{L}_2)|A|-o(|A|).$$

The source presents this as a stronger bound, modeled on the Krachun--Petrov conjecture, and explains that it would imply the paper's main theorem in the coprime case. Its generalization to more than two transformations is stated to be unclear.