Let $F:\cal C\to D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky Locally presentable and accessible categories, there exist arbitrarily large regular cardinals $\lambda$ such that $F$ preserves $\lambda$-presentable objects. It is tempting to expect that $F$ should preserve $\lambda$-presentable objects for all sufficiently large $\lambda$, but that is not what the theorem says. However, I do not know a counterexample showing that the stronger claim fails. (For instance, this question asks about this property when $F$ is the pullback functor, and has no answer yet in the general case.)
What is an example of an accessible functor $F$ between locally presentable categories for which there exist arbitrarily large regular cardinals $\mu$ such that $F$ does not preserve $\mu$-presentable objects?