I listened to the sound clip and it sounds exactly like string noise behind the nut. Try wrapping a piece of foam or cloth around them and see if the noise goes away. You can get a fancy FretWrap and place it behind the nut.
As a Floyd Rose user, I can confirm that spring noise is also a thing. If you’ve got trem systems on your guitars, it’s a good idea to put some foam in the rear cavity to cut down on spring vibrations. Floyd Rose also makes a type of coated spring which helps out some.
A quick fix is that you can find the frequency of the noise and attenuate it with an EQ block in the QC, but there’s no guarantee that it will be the same across multiple guitars, and it will affect your overall tone no matter what. I would suggest turning your gain down a bit (in addition to fixing string/spring noises in the guitars).