You are confusing fact and perception thereof. Fact Can and most Do exists independent of people perception or interpretation, e.g. natural facts, like the average distance of an hydrogen atom at rest state. We hold the press to try to report this. But then somebody come and VERY obviously tell an utter completely wrong headed lie. This is not anymore a question of perception, interpretation, or fact. It is a question of complete anti factual lie.
If you associate that with interpretation or second order belief , then you may as well use doublespeak and war is peace.
I tried to distinguish between demonstrable (as you term it "natural") facts and socially constructed truths. As a materialist, of course I agree there are facts which don't care one whit about my personal opinion. However, this is not what we are talking about.
Let me put it this way. Suppose I have an electronic scale I use in my kitchen to weigh out ingredients. Unbeknownst to me, it has a flaw. When the humidity in my house is high, the scale reads as much as 5% off the true, factual weight. The error is not obvious to me because the few times I've checked my scale, it performed well enough. It is also not obvious because the ingredients I measure, and the dishes I make from them, are not greatly affected by the error.
Now, because we have this outside, omniscient perspective, we know what's going on. But if it were just me and my scale, I would never know. I would state, with high confidence, that whatever weight it showed was correct.
Switch to the current argument. I am presented with two sets of evidence, two stories about what the facts are. I assume that at most, only one story can be true (they both might be wrong). I cannot directly check for myself but must rely on others. Whom should I trust and why?
Here are some choices:
1) Believe the narrative which most closely matches my existing bias
2) Try to sort narratives and compare the evidence offered up
3) Check with other people I trust and see what they have to say
4) Ignore the whole thing as not deserving any real work on my part
But underlying all this is the possibility of a defective scale - that is, human beings who think they are correct but aren't. The theoretical existence of indisputable facts does not mean that I have access to them, nor that I will know them when they punch me in the nose.