Conlon–Lim's Brunn–Minkowski conjecture for discrete linear sumsets

Let $d,k4in\mathbb{N}$ and let $\mathcal{L}_1,\dots,\mathcal{L}_k\in\operatorname{Mat}_d(\mathbb{Z})$ be irreducible and coprime. Let $A\subseteq\mathbb{Z}^d$ be finite and non-empty. Here, irreducibility means that there are no non-trivial subspaces $U,V\subseteq\mathbb{Q}^d$ of the same dimension such that $\mathcal{L}_i(U)\subseteq V$ for every $1\leq i\leq k$, while coprimality means that no $\mathcal{P},\mathcal{R}\in\operatorname{GL}_d(\mathbb{Q})$ satisfy $0<|\det(\mathcal{P})\det(\mathcal{R})|<1$ and $\mathcal{P}\mathcal{L}_1\mathcal{R},\dots,\mathcal{P}\mathcal{L}_k\mathcal{R}\in\operatorname{Mat}_d(\mathbb{Z})$. Conlon–Lim's conjecture. One has

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

This conjecture proposes a Brunn–Minkowski-type lower bound for discrete sumsets generated by irreducible, coprime integer linear maps; the source presents it as a conjecture from Conlon–Lim and gives no resolution.