Ground loop isolation means.... no ground reference between components!
If you ground both ends, you have re-established the ground and are now NOT ground loop isolated.
That is why you do not ground both ends.

In normal use, you do not ground either end. No shielded cable and no shielded plugs, that can carry a ground to the other side.

Some audio equipment requires a ground reference in order to work (very rare these days). That is what the ground terminals are for. Whichever equipment is requiring the ground reference, you run a short ground wire from that ground terminal to the equipment chassis. And NO where else.

True baluns have a small transformer in them. Make sure the little black box is not up against other electronic equipment. Those coils are antennas as well and will pick up hum from sources very close by.