Ruzsa's discrete Brunn–Minkowski conjecture

Let $k\in\mathbb{N}$ and let $\epsilon$ be positive. For finite sets $A,B\subset\mathbb{Z}^k$, say that $B$ is not covered by $n$ parallel hyperplanes when no collection of $n$ parallel hyperplanes contains $B$. Ruzsa's discrete Brunn–Minkowski conjecture. For every $k\in\mathbb{N}$ and $\epsilon$ there exists $n_{k,\epsilon}$ such that, whenever $A,B\subset\mathbb{Z}^k$ and $B$ is not covered by $n_{k,\epsilon}$ parallel hyperplanes, one has

$$|A+B|^{1/k}\geq |A|^{1/k}+(1-\epsilon)|B|^{1/k}.$$

This conjecture seeks a discrete analogue of the Brunn–Minkowski inequality under the natural minimal non-degeneracy assumption proposed by Ruzsa. Earlier results obtain the desired asymptotic bound either under stronger density assumptions or give substantially weaker bounds under full-dimensionality assumptions.