I am trying again on 40 meters with the ATX indoors. It seems to make a big difference whether the radial is just thrown down on the floor or it is deployed to its full length. In the latter case SWR comes down to 1.1 or "1 bar" on the FT-817, which is somehow better than what I had outside, while in the first case I just couldn't get the SWR below "4 bars".
One bad thing about the ATX-1080 though... I bought the Mk 2 version which includes an BNC plug and a 90 degree PL plug. So far, I have only been using the PL plug because it can be mounted directly on the back of the tuner or radio. Now it turns out that I can not mount it in a perfectly vertical position but I have to tilt it 5 degrees if I want to tighten it. The small "things" on the plug which are supposed to lock on the female plug are tilted. Well, this can of course be a coincidence, but I tried both my tuner, the FT-817 and the IC-765; none of them can hold the antenna in a vertical position :o)
Furtunately, I can turn the socket of my tuner but I can not do that on the FT-817.