This only matters if it's important to you that everybody believe the same truths, which I would suggest shouldn't be important to you. Some people will believe one thing, another group will believe a different thing, and those disagreements can't always be reasoned away. Which should be a perfectly fine outcome, it shouldn't cause you any distress that people believe things that you think aren't true, and vice versa.
Trying to boil this down to the quality of the faith is also a rather immature response. To frame things this way isn't to accept that there are different viewpoints other than your own, it's just to assert a claim that your viewpoints are correct, and that while other view points might exist, they are wrong. Your assessment of what is a good faith or what is a bad faith argument likely has little to do with the quality of the arguments involved, and instead will somehow miraculously align with your own world view at a rather implausible rate.
If you want to argue with people in public, the only thing you should really be concerned about is stating your best case. If you do that then you've achieved the only mature goals that you could possibly attach to public arguing, and whether people are convinced by it or not is up to them.
Trying to boil this down to the quality of the faith is also a rather immature response. To frame things this way isn't to accept that there are different viewpoints other than your own, it's just to assert a claim that your viewpoints are correct, and that while other view points might exist, they are wrong. Your assessment of what is a good faith or what is a bad faith argument likely has little to do with the quality of the arguments involved, and instead will somehow miraculously align with your own world view at a rather implausible rate.
If you want to argue with people in public, the only thing you should really be concerned about is stating your best case. If you do that then you've achieved the only mature goals that you could possibly attach to public arguing, and whether people are convinced by it or not is up to them.