On Ruzsa's discrete Brunn-Minkowski conjecture

We prove a conjecture by Ruzsa from 2006 on a discrete version of the Brunn-Minkowski inequality, stating that for any $A,B\subset\mathbb{Z}^k$ and $\epsilon>0$ with $B$ not contained in $n_{k,\epsilon}$ parallel hyperplanes we have $|A+B|^{1/k}\geq |A|^{1/k}+\left(1-\epsilon\right)|B|^{1/k}$.