The periodic tiling conjecture for regions

A periodic set $\Gamma\subset\mathbb{R}^n$ is one of the form

$$\Gamma=L+R\mathbb{Z}^n,$$

where $R$ is an invertible matrix and $L\subset\mathbb{R}^n$ is finite. A region is a closed subset of $\mathbb{R}^n$ equal to the closure of its interior, with finite positive Lebesgue measure and boundary of measure zero. A region tiles by translations if its translates partition $\mathbb{R}^n$ up to measure zero.

Periodic tiling conjecture. If a region $\Omega\subset\mathbb{R}^n$ tiles $\mathbb{R}^n$ by translations, then it has a periodic tiling.

The conjecture asserts that translational tilings of sufficiently regular finite-measure regions always admit periodic translation sets. The supplied text gives no resolution status.