When it is said that neutrinos don't interact with matter, it means, that neutrinos don't usually strike other atomic particles. But there are neutrino detectors set up to find those very rare occasions when collisions do occur, because interesting things seem to happen.
The moon doesn't touch the Earth, but it has an effect on tides. What if a neutrino passing near an unstable nucleus causes it break down faster?
To be sure, I'm not convinced either. But definitely want to see more, controlled testing.
