Some aspects of locally presentable category theory depend on the assumptions of set theory. Namely, a large cardinal axiom known as the Vopenka principle implies several interesting properties of such categories (and some of them equivalent). For example, this implies that each category complete with a dense subcategory is presentable locally.

Now let $ mathcal {U} $ to be a universe of set theory such as a Grothendieck universe. Then we can consider the locally presentable theory $ mathcal {U} $-categories, where, basically, we replace all small sets by $ mathcal {U} $-small sets. There is a pre-print that studies such categories, but it does not consider properties that depend on the Vopenka principle. So, are these properties still true for locally presentable $ mathcal {U} $-categories or do they still depend on hypotheses of set theory?