I can't speak for DisplayOut but I use TVOut2 which seems to work fine for me. I'm evenable to use it over AirPlay with AppleTV3 and initially when you open the app it states that you need to disconnect the adapter but once you browse to a channel, it displays on the TV.
On a side now - When I try to use the Digital AV HDMI Adapter on my iPad 5.0.1, I get a message that states "Accessory Not Supported: Please Update to Latest iOS." - Anyone ever see this?