The conspiracy that Fox missed (well, they do miss a lot) is called by its adherents "Alternative 3". Its promoters are shadowy individuals like Bill Cooper, a formal US Naval Briefing Team member with access to state secrets, who claims that an advanced American space presence is a reality. In Behold a Pale Horse, he reports that "A moon base, Luna, was photographed by the Lunar Orbiter and filmed by Apollo astronauts... I can say that 'Alternative 003' (a British TV documentary on this subject) is at least 70% true from my own knowledge and the knowledge of my sources."
Dr. Thomas and his team at Cornell University have tried to peer into the mysterious interior of the weird little Space egg that is Methone. They started out with the hypothesis that Saturn's relentless strong gravity pulls the little moon into an elongated shape, just like Earth's own large Moon raises ocean tides on our own planet. Then the team went on to calculate how dense the little moon would have to be for its own gravity to counteract those intense tidal forces and create its strange egg-shape.
Now of course this doesn't mean that you shouldn't let go of something during the waxing moon, or that you can't sign up for something new in the Waning moon. It does mean that if you have plans for this kind of work, including the tidal pull of the different moon phases in your plan, will support your actions in the same way that swimming with or against the tide makes your swim easier or more difficult.