When the facts change, loyalty to the old conclusion becomes stubbornness.

Holding a position after the grounds for it have gone is not integrity, though it can be defended in the language of integrity. Consistency is a virtue about principles rather than about conclusions; a conclusion is supposed to move when its supports move. The reluctance is usually about not wanting to have been wrong in public.

When new information undercuts something you have argued for, say so explicitly rather than quietly adjusting. It costs a moment of discomfort and it buys you a great deal of credibility, because people notice who updates and who does not.