SCHWEITZER 1986 Problem 9

Consider a latticelike packing of translates of a convex region $K$ . Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a latticelike packing of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]