That's for Red Bull to decide, they may see whatever PR damage this causes to be worth less than the damage of him leaving on short notice without proper hand over/ training his successor. Overall Red Bull are great PR machine with all the things they get involved in. Losing some F1 fans and maintaining their performance is likely worth more than small (relatively speaking) loss of fan base, which they can recover by continuous success in the future.
If it was just the F1 team, I think they'd probably stick it out. But the F1 teams is a small part of the whole operation and the last thing Red Bull proper want is the F1 team harming the main company. Ultimately the F1 team is marketing to them, and if marketing is generating negative publicity...
I think it's obvious Horner has done a great job as team principal, and they've benefited an enormous amount from sticking with a stable setup, but I don't think he's irreplaceable. They will still have Newey and Verstappen, and all the other engineers and so on that are making the team such a success if he leaves. It seems rather unlikely that there isn't someone else who could step in and lead the team to continued victories.