It is possible to have a low positive 2 weesk after infection. By 3 weeks after infection, 50% of people who are going to be positive will be positive, so absolutely, someone could have a low positive index value 2 weeks after infection.
Most often, numbers will increase after a new infection, correct.
You can consider this a false positive if you wish. I’m afraid I can’t give you a 100% certainty here.