In your thought experiment you make the weight difference so small that otherwise insignificant factors (e.g. friction) take over. Go the opposite direction and make the weight difference enormous.
Assuming no friction and given a fast tug of the rope I still don't think the near equal weight would hold still or sink while the other near equal weight flew up quickly.
With the large weight difference, I think the required speed of pulling the rope to make them both rise just becomes impractically fast.
Imagine the system in space - all the weights would move up, so they all have an upward force applied to them from the rope tugging. It's just a matter of pulling so fast that that upward force overcomes gravity.
