In a locally presentable category $ cal E $, there are arbitrarily large regular cardinals $ lambda $ such as $ lambda $-presentable (a.k.a. $ lambda $-compact) are closed under withdrawals. Namely, the withdrawal functor $ { cal E} ^ {( to leftarrow)} to cal E $ is a right assistant, so accessible. So, he keeps $ lambda $-presentable objects for arbitrarily large $ lambda $so just check that the $ lambda $objects -presentable in $ { cal E} ^ {( to leftarrow)} $ are the ones that are punctual in $ cal E $. (A version of this argument is given in this answer in the case of finished products.)

Of course, "arbitrarily big" means that for any cardinal $ mu $ there is a regular cardinal $ lambda> mu $ with this property. A stronger statement would be that this is true for *all big enough* regular cardinals $ lambda $, that is, invert the quantifiers and say that there is a $ mu $ such as all regular cardinals $ lambda> mu $ have this property $ lambda $-The presentable objects are closed under withdrawals). Is this stronger affirmation true?

Note that this is certainly not true *all* regular cardinals $ lambda $ have this property; counterexamples can be found here.