With default opamps I got the same hum. Like I said before, rearranging power transformer resolves the issue. I have reasons to believe that all E1s are like that because they all share the same trafo part no., so all you need is AKG K550 and a dead silent room (at 2AM in the morning) to do the testing. :) You will not hear hum with any kind of headphone, even not with all IEMs, so try it out with K550 please.
Feel free to check the screenshots here: http://www.homemedia.fr/tests/i/mesures/asus-essence-one-rmaa/ASUS%20Xonar%20Essence%20One%2096%20khz.htm and http://archimago.blogspot.ro/2013/02/measurement-asus-xonar-essence-one.html.You can see the hum I'm speaking about, especially 2'nd and 3'rd harmonic (around 200 Hz and 300 Hz). DACs having external PSU doesn't have this hum at all...just pure white noise maybe, but not hum.
My Beyerdynamic DT770 PRO are among several HP's that are references for low frequency reproduction, down to infrasonic. Posted links show that there are no peaks above -100 db, and that is unhearable at low frequencies. Your situation must be much worse than that.
Here are two of my measurements taken two years ago during extensive swapping.
This is loopback measurement of my Terratec audio card in standard PC.
This is E1 measurement with LT1028+OPA627AU amps.
As you can see there is only slight contribution to Hum and Noise by E1.
E1 is obviously much better than my measuring audio card.
Now I use 6x Muses 01 opamps and that is the best setup (tried Muses 02 also).
Have you tried to move E1 away of the other possible sources of interference.
I have noticed that mobile phone can make noises in E1 if placed less than 1 meter close.