The AKM4137EQ is an asynchronous sample rate converter. It appears as though they will use this to resample any input data rate.
I found
the manual. For a $3000 device you would hope for something a bit better. It is no better than I would expect in a $100 device. Anyway, it appears as if there is a lot of setup options. All the different filters are there. Some EQ options as well, and software control of the output separately for for 3.5mm and 4.4mm outputs. One can set different gains. +14.4dBu(High Gain) -5.6dBu(Low Gain) I suspect the settings were different for the two outputs, and this explains the difference measured. The difference is 20dB, and this is exactly what Amir measured. So the unbalanced output performance is probably the same as the balanced so long as you set the gains. The instructions give no clue as to how this might be effected.
There are ATE (which I assume are pre-canned equalisation) settings. "Brighter" "Sweet" "Dental".. Dental??? Puts one's teeth on edge perhaps? Again, zero clue in the manual. At this price point that is unforgivable. I would expect a professionally written, expansive and complete manual, and a hard copy in leather binding.