This actually feels like a variant on overjustification. In the absence of evidence, people are content with their belief on whatever intuitive basis led them to adopt it in the first place. Provided weak evidence from an external source, they consciously reach for a better reason to believe and don't find one.
Article: Weak supporting evidence can undermine belief in an outcome
Paper: When good evidence goes bad: The weak evidence effect in judgment and decision-making
Abstract: