Cool question.
Fighting over a belief is rarely worth it. When you identify too deeply with a belief, you take any criticism of that belief as a personal attack. That's a recipe for disaster. It's fine to argue your points, but separate it from yourself and if it's obvious you're not getting anywhere, just drop it. The goal should be taking in new information, not necessarily "winning".
One of the few conflicts I'll escalate over are personal attacks of character. If somebody accuses me of being dishonest and I'm not, that's not something I'm dropping. I'd sooner end that relationship than acquiesce the point.