I was listening directly out of the Sony. Maybe it was the amp? I'll try it as a source if I remember to.
I think you may be right about the balanced out making the most difference, going by some back-and-forth listening between the AK240 and other DAPs and whatnot tonight. It is inclining me even more to my thought that, other features and the rest of the design notwithstanding, it is best to be considering this DAP only if one intends to use the balanced output with a pair of the best IEMs or headphones as desired, obvious as that sounds.
Please try Your Sony ZX 1 through digital line out use Sony Lod plus mini to mini pair to Your portable amp
You Will Love THE SQ , it is better than through HP Out