A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider context, a solution to this problem would provide another proof that Artinian local rings whose maximal ideal is principal are principal ideal rings by shifting from Artinianness to Noetherianness instead of exploiting the nilpotence of the maximal ideal.
I'm tagging this commutative-rings because those are the only ones I really care about, but a noncommutative example would be just as decisive.