No. They do not know the belief isn’t true. “Knowledge” A belief held to such a degree that it would be life-altering were it to be demonstrated wrong." If you are going to discuss Knowledge with me, this is the definition. If you have a better one, let’s hear it. All things point to their knowledge being true. “Look at the trees.”
A Gnostic atheist does present contrary evidence. The best argument I have seen them use is the argument from divine hiddenness. The evidence is actually stronger than the evidence for a god.
Absence of evidence is evidence when that evidence would be naturally expected. If I told you I had a dead body in the trunk of my car, we could go out to the car and look . If there was no body in the car I could tell you someone had taken it. So we look for hair. Again someone cleaned the car. So we look for indents in the carpet. We find none. We look for clothing material, we find none. We look for body fluids of any kind. We find none. We look for fingerprints. We find none. All of this stacks up to evidence that there was never a body in the car. It can not be 100% proof, but it will stand up in any court and it is not hearsay. We have 2,000 years of this sort of evidence in addition to all the failed man-made gods. It is special pleading to make an exception for one more with no more evidence supporting it than the ones that came before. A Gnostic atheist ‘Does’ make the assertion and he or she ‘Does’ make the argument.
Now, with that said, I regard all of these as positions, not as people. Sometimes I am an agnostic atheist, and sometimes I am an anti-theist or Gnostic atheist. It depends on the god being presented. I can disprove many gods that are presented to me. (That does not mean people can not identify as either.) I’m an atheist and I make the rounds depending on the god being discussed.
I most certainly can produce evidence against a god. Which one do you have in mind? Please clearly identify your god. And, how you know it is a god.