Funny you should ask. At last weekend's Bay Area meet, this was the comparison going on at my PC. Pico plugged into one USB port, Headroom Portable Desktop plugged into another.
Every person I talked to came away stating the Pico sounded significantly better, FWIW. I didn't listen to the Headroom too much, but it certainly had a darker sound, and was a bit more blurred/muddled sounding, compared to what I am used to from the Pico. Granted, it did have more volume reserves, and drove the K340 a bit better than the Pico, but considering it has a battery larger than the entire Pico, it probably should have.