You can disprove this idea even in a simple thought experiment, you don't even need an actual experiment. If I take 2 large stones, connect them with a string so it's 1 object, will that object fall twice as fast than the 2 stones fall separately?
Working under the theory that large dense things fall faster than small things, you'd expect that the large stone would drag the small stone. Much like if you attach a piece of wood to a rock with a rope and throw it in water, the rock will drag the wood down.
You'd probably argue that even though they are connected with a string they are still 2 things and there's a density gap between the two of them.
Now, if you try and figure out what makes a thing a thing, it starts breaking down. But if you've already have the working theory, then making an explanation isn't terribly hard.
That will depend on the length of the rope, once aerodynamic effects are taken into account