In a locally presentable category $\cal E$, there are arbitrarily large regular cardinals $\lambda$ such that the $\lambda$-presentable (a.k.a. $\lambda$-compact) objects are closed under pullbacks. Namely, the pullback functor ${\cal E}^{(\to\leftarrow)}\to \cal E$ is a right adjoint, hence accesible. Thus it preserves $\lambda$-presentable objects for arbitrarily large $\lambda$, so it's enough to check that the $\lambda$-presentable objects in ${\cal E}^{(\to\leftarrow)}$ are those that are pointwise so in $\cal E$. (A version of this argument is given in this answer in the case of finite products.)
Of course "arbitrarily large" means that for any cardinal $\mu$ there exists a regular cardinal $\lambda>\mu$ with this property. A stronger claim would be that this is true for all sufficiently large regular cardinals $\lambda$, i.e. to reverse the quantifiers and say there exists a $\mu$ such that all regular cardinals $\lambda>\mu$ have this property (that $\lambda$-presentable objects are closed under pullbacks). Is this stronger claim true?
Note that it is certainly not true that all regular cardinals $\lambda$ have this property; counterexamples can be found here.
