Before revision:  Our results  imply that the class of 1-perfectly orientable graphs is (tw,\omega)-bounded, answering a question of Bresar, Hartinger, Kos  and Milanic from 2018.
After revision:  Our results imply that the class of 1-perfectly orientable graphs is (tw,\omega)-bounded, leading to linear-time algorithms for k-coloring 1-perfectly orientable graphs for every fixed k. This answers a question of Bresar, Hartinger, Kos and Milanic from 2018.