Every so often a mistake is made where the price to pay for correcting it (at best) or apologizing for it (at least) is just too high.
The term used in those situations is usually "dearly." Dearly is the general cost of being wrong about something important.
I have been wrong about some very important things, and I am paying dearly.
But I am not about to apologize. Call it obstinate, call it what you will. But do not call requesting an apologia.