I have exactly the same problem with my Unpowered Kemper Rack. Bought it used a couple of months ago and until yesterday I was using it with no issue at all. It is a 2017 model.
Yesterday I opened my Kemper and I am getting the same noise as yours. I've tried different guitars, different audio interfaces, plugged directly into Kemper, nothing.
I've also tried different plugs and places in the house to make sure that the problem was not due to interference from another electric device.
And all of this exactly like you "with nothing plugged in except the power lead and a set of headphones".
I am going to open a support ticket (obviously) but I find it strange that we are experiencing the same issue at the same time.
P.S. @akashic777 I would be really interested to know if your matter is solved and how!