The Form 4 which I just got back from ATF, calls the device a "silencer". I didn't fill it out, the retailer who sold me the device did.
For reference, it took five weeks to get the local LEO (sheriff Joe Arpaio) to sign the application. He did send me a written apology for the delay. The trip to the ATF took another 2 months.
All-in-all, its a gadget and an expensive one at that--$425 for the "device" and $200 for the stamp. As noted above, it would have legitimate purpose(s) short of clandestine use, but I got it as a novelty, and to "experience" the paperwork required. Given that the paperwork has to be repeated for every controlled item one would think that the market for supressors, SBRs, etc would be limited, but there are shops locally that specialize in that kind of item, so guess not. For the guys going after the full mgs, the waiting period is probably not a big deal.
FWIW, the supressor (Sparrow model, by Silencer Co.) makes the pistol sound like a pneumatic stapler using subsonic ammo. I thought the first shot was a squib until I saw the hole in the target.