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, answering a question of Bresar, Hartinger, Kos and Milanic from 2018, and one of Beisegel, Chudnovsky, Gurvich, Milani\v{c