Well after bringing the LCD-2 up to HE-6 territory, the loft that houses the "high-powered speaker rig", I can confidently say that the best of these two headphones is more than ever the one that's currently on my head. If the hallmark of a great can is the overwhelming desire to "turn it up" at all times, and the desire to listen to the dustiest material in one's collection with the freshest of ears, then I think that we have a winner with both of these planars.
If pressed I may admit that the HE-6 seems to be more technically proficient, or that the LCD-2 has a bit more warmth in its response. But that isn't to say that the LCD-2 lack rhythm, or that the HE-6 is anything less than musical in its presentation. In some ways I am glad that they have such different amping requirements, as having them together on the same rig all of the time could present me with too much of a dilemma.
These are both such radically perfect headphones in their sonic presentation that amping requirements could be the only real point of technical separation, meaning if you have a great amp that doesn't quite meet the power requirements of the HE-6 it could still be a great match for the LCD-2. On the other hand, if you are already set up for a K1000 and want to upgrade, the HE-6 could be perfect for you but I still wouldn't rule the LCD-2 out!
Overall I would say that the combination of being easier to drive and the $250 price difference makes the argument quite compelling on the side of the LCD-2. Subjectively, when you add the natural wood housing it wins aesthetically for me as well, but some could easily feel differently and prefer the piano-gloss look of the HE-6.
Either headphone can reach their potential off the speaker terminals of a decent vintage receiver so if you are building a system around either one, amplifier cost shouldn't be a factor. Another subjective thing for myself is the fact that the LCD-2 has such an all-American pedigree, global competitiveness being nothing to sneeze at these days.
So the answer to the OP continues to be no for this listener, and happily so