In this case, "beliefs" means "ethical opinions."

No ethical opinion seems liable to either objective proof or disproof.

So the dilemma is unlikely to arise.

For the present purpose, a "fact" is something that is demonstrably true or has or has had verified actual existence.

It's fact that in counting separate items, 2 and 2 equal four. Vampire bats are a fact.

Beyond that, we're on our own.