Positroids are 3-colorable

We show that every positroid of rank $r\ge 3$ has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J.
Ne\v{s}et\v{r}il, R.\ Nickel, and W.~Hochst"{a}ttler, this shows that every orientation of a positroid is 3-colorable.