Fake information is not false consciousness. False consciousness would be if people believe that something like an unregulated market place benefits them, because it makes everyone richer. Kinda like thinking of tax cuts to wealthy as good, despite lots of the things being cut actually being services that benefit you, and the unlikelihood of those cuts giving any sort of economic boon to normal people.

Don't get me wrong, fake information is incredibly frustrating and difficult to deal with, but there is a distinction that needs to be made here. People's minds are very difficult to change, but rational individuals should be able to hear your evidence, and all you can do then is make sure that the evidence is sound, the arguments drawn from it make no large mental leaps, and the conclusions are reasonable. In the case of irrational people, that is a completely different story, but identifying and disproving their fears is a valid strategy, or perhaps showing them that things they stand for can work within a different fram of mind, or attempting to show contradictions in their thinking. Critical thinking is not something that is easily fostered, but it is safe enough to say that humility and respect are important steps on the way. One thing to never do, is to tell them that they suffer from false consciousness, because anyone with a sufficient understanding of the concept will accuse you of the same thing.