Epic Games explains in its official Twitter account that Apple will extend this registration indefinitely. By this they mean that users could continue to use this option to register but Epic also advise the change of credentials to their users and is that at some point Apple support may close. A few hours after stating that Apple would close the option to register with "Sign in with Apple" in Epic Games accounts it seems that there is a small truce in this sense.
The tweet edited and updated by Epic is:
UPDATE: Apple previously stated they would terminate “Sign In with Apple” support for Epic Games accounts after Sept 11, 2020, but today provided an indefinite extension. We still recommend you prepare your accounts now for “Sign In with Apple” removal. https://t.co/hUDCKcajGo
- Epic Games Store (@EpicGames) September 10, 2020
Apple itself has not commented on this issue and that is why there is some uncertainty in this regard. What Epic Games recommends is that you access your data on the website itself and modify your credentials to avoid access problems In the near future, the closing of this option is confirmed.
It does not seem that this conflict between the two companies has a happy ending, so it is advisable to have everything well tied to avoid possible problems on all fronts. Of course it is a case of those that brings a tail and neither of them would be willing to give in at this time so the user has to avoid problems of any kind.
