Bukh's linear-transformation sumset conjecture

Let $\mathscr{L}_1,\dots,\mathscr{L}_k$ be linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ that do not share a non-trivial invariant subspace and satisfy

$$\mathscr{L}_1(\mathbb{Z}^d)+\dots+\mathscr{L}_k(\mathbb{Z}^d)=\mathbb{Z}^d.$$

Bukh's conjecture. For any $A\subseteq\mathbb{Z}^d$, one has

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

This conjecture concerns sharp asymptotic lower bounds for sums of images of an integer set under linear transformations, and is related to estimates for algebraic dilations. Its resolution status is not specified in the supplied text.