Absolutely not true.

Assuming that you would use the same bow set at the same poundage and draw length to do a test........

A heavier arrow will outpenetrate a lighter arrow given that they are fletched the same, the same diameters, and the same length. The heavier arrow will absorbe more of the bows available energy than the lighter arrow will. That's why when you shoot a light arrow the bow is noisier because the bow did not transfer as much available energy to the arrow as it would a heavier arrow. The longer an arrow stays on the string the more energy it gets from the bows available stored energy.

The heavy arrow will have a tad bit more KE and a good bit more momentum. It is harder to stop a heavy slow moving projectile than it is to stop a light but fast projectile.

Of course you can go too heavy or too light. There is a point of diminished returns when talking about arrow weight.