Halin's grid theorem for digraphs

Halin showed that every thick end of every graph contains an infinite grid. We extend Halin’s theorem to digraphs. More precisely, we show that for every infinite family $\mathcal{R}$ of disjoint equivalent out-rays there is a grid whose vertical rays are contained in $\mathcal{R}$. Furthermore, we obtain similar results for in-rays and necklaces.