Koszulity for semi-infinite highest weight categories

We show that any upper finite or essentially finite highest weight category where the standard objects have linear projective resolutions and the costandard objects have linear injective resolutions is Koszul. This extends the result of Agoston, Dlab, and Lukacs to the case of infinite highest weight categories. We apply this result to Khovanov algebras and representations of classical Deligne categories and show that these are Koszul.