Boundedness of geometrically integral $\epsilon$-klt del Pezzo surfaces

We prove the remaining positive characteristic cases of the Borisov-Alexeev-Borisov conjecture for geometrically integral $\epsilon$-klt del Pezzo surfaces over arbitrary fields. Bernasconi and Martin established boundedness outside characteristics $2$, $3$, and $5$, and reduced the remaining cases to a uniform bound for the irregularity. We prove this bound, thereby completing the boundedness theorem in all positive characteristics.