@td47: Good morning, thanks for your input here. I checked already out that function before and the function Auto On / Off was always activated. It also works for all other inputs, besides for the "Port"-input. Apparently, here happens exactly what's quoted in the description you copied. The HDMI-ARC adapter seems to get always power from it's USB plug and therefore the Auto On / Off function of the speakers is not working.
However, this goes a bit in a different direction that I was heading to. In my setup, I don't want to rely on the Auto On / Off function of the speakers themselves- I wanted to use the "Magic Remote" and HDMI ARC abilities of my LG TV. There, I would have expected that the TV sends a "Turn Off"-signal via HDMI ARC to the speakers and tell them to shut down too. Same when turning on the TV. And what's still mysterious to me, that it does work during the configuration mode. I can turn on and off the speakers by pressing a button on the TV during the test out phase of the HDMI ARC configuration. However, after completing that configuration, the only function which functions is adjusting the speakers volume with the Magic Remote. Unfortunately, the synchronization of the Power (even with the function activated) remains unfunctional.
Looking forward to any other suggestion.