I'm not sure they do. It's an extremely counterintuitive claim that would need to be justified, and while the author does cite (their own) paper, it sounds like the model they came up with is highly parameterised and not particularly physically validated. If it's really the case that contrails reflect more heat down than up (unlike what the scientific consensus says is true for regular clouds), then there should be an explanation for what contrail-specific factor causes this, not just "here's a pile of math equations that say it doesn't, don't ask where we got the parameters to fill them out from".
