We all have beliefs and opinions that will likely remain unchanged no matter what counter-evidence is brought to our attention. This is actually typical of all humans, since our politics, faith, values, and views are shaped by psychological and social conditions that are largely outside of our control (and usually unnoticed in their influence).
Yet every one of us will claim that our beliefs are based on sound reasoning and facts — in contrast to our opponents, of course. In that case, we should ask ourselves the following: what would it take for me to accept my opponent’s beliefs? What sort of proof would I need to discard my deeply held views?
If you can’t find any reason why you should think differently, then in essence you’re admitting that your views are purely visceral rather than evidence-based. Either change the basis of what you believe — i.e. try to find evidence for it, and discard it in the absence of said evidence — or admit that your beliefs have nothing to do with reality or rationality, but are instead the result of unthinking emotional or psychological attachment.
If we’re going to take up a belief or opinion based on “gut instinct,” faith, or whatever else you want to call it, then we might as well be honest, recognize it, and not hold it against others if they don’t see eye to eye with our views (after all, if said belief is based on personal feelings, rather than something objectively measurable, then you can’t expect everyone else to agree).
Also, there’s no harm in saying “I don’t know” or “I believe this based only on what I know.” It’s honest and it represents a fact of life: not everything is knowable to everyone.
To the best of my ability I try to hold myself to these standards. Otherwise, I leave it to others to call me out.