When both sides have "proof" and "know" they are right, the means of conflict resolution are somewhat slim, either one side has to voluntarily concede to the other, or there is the tried and failed method of gathering your gang and winning by numbers. You could agree to disagree but that doesn't really solve the question of who is actually right and who is wrong, who's knowledge is validated and who's knowledge is a load of candy fluff. You could try scientific inquiry, but either side can complain that the tests weren't conducted quite right, or claim that the scientists were bribed or bought out, or they could simply refuse to believe that it can really be testable, or throw out any one of dozens of arguments to try and trip up their opposition, bringing us back to square one on the subject, How does anyone win in this situation? As far as I can tell, There is none. Not even if one side concedes, because then the two of them are going to meet another person who disagrees with them. And another after that. And another after that. And the conflict will just rage on and on and on and on and ad infinitum. Even if a large number of people were convinced that one thing was correct, does that mean it is right? History contains evidence that perhaps masses don't make absolute proof. There have been popular beliefs that have been abandoned by later generations?
Can the battle for absolute knowledge be won then? That is a dang good question. I think not. I think that the best we can do is stake our position on a subject and stick with it, and agree to disagree if no one is going to concede. If we are wrong, we are wrong together. That is the nature of the beast known as knowledge. Knowledge, as I stated above, is fickle. We can try to predict things, and we can even be pretty good at it, but knowledge about the future is non existent, the best I'm afraid we can do is prediction. For example, you can predict that the sun will come up tomorrow, and for all practical purposes, you are probably right, the sun will probably come up tomorrow, but the fact is that other possibilities could exist, There could be a freakishly fast moving black hole that is approaching the sun from the exact opposite side of the sun where we might not be able to see, and in the night the sun could be absorbed by this black hole. I will say, it is not likely that this will happen, but it could, and therefore I can say I bet the sun will come up tomorrow, but I don't absolutely know it will happen.
Should this battle be won? For pragmatic purposes, No. Just as having genetic diversity among a species better grantees the species existence as a whole. Having a diversity of opinion and belief or "knowledge" helps to keep our social and political systems in a state of balance, and it creates conditions for the existence and development of a wider variety of life styles, skills, problem solutions, etc that may be needed to deal with problems that might arise in the world.
Anyway. So it is my belief that we can have strong enough belief that we can rely upon it like knowledge, but that absolute irrefutable knowledge can not be obtained. (Ironically, and I have seen this happen, there are other individuals who do not believe this basic premise on knowledge outlined here, and they do believe that some sort of absolute irrefutable knowledge exists and that, of course, they possess it. So as I am writing this, I know that there are others out there who would disagree with me even on this point of basic philosophy. I maintain that it is their right to believe the way they do, and I will not dispute it.)