After some reverse engineering I already now how it works, however I haven't measured anything directly on the radio side yet.
In twin mode, on the control side, locally to me, I can see a "PS;" command coming on the serial line of my K3/0 (mini) in every 2-3 seconds. This is a question from the radio (or the local RRC box, se later) about the requested power state of the control side. When we press the button on the K3/0, it starts sending "PS1;" messages repeteadly until the K3/0 is turned off, then it sends a "PS0;". Also, if the sequence of "PS1;"'s stops suddenly because of any problem (power is removed from the K3/0, internet connection breaks, etc.), the radio will turn off itself after some seconds, too.
I don't know yet what causes the radio to wait for a K3/0 to be attached initially, it might be the line of the K3 radio's back that needs a pulse before you can turn it on remotely when already powered. There is a state when the radio is not fully on, but its display is lit - if you hold that line high without releasing. I observed that the RRC's provide some more sophisticated interfacing, so you are not necessarily getting the physically same PS; questions from the radio, but they come out of the control RRC, and you can respond to them anyway.
And what I'm trying to say is: when I hadn't have a K3/0 yet, I just connected the appropriate port of the control RRC communicating normally with the K3/0, and got a serial terminal program to send PS1;'s to the control RRC blindly in every 3-5 hundred milliseconds to turn on and against turning off the K3 itself, then I used the whole system through the also available CAT port going through the RRC's (a port programmed to be the CAT port on the control side - it can be one of the serial ports or a USB virtual serial port in the Windows as well speaking of desktop PC's).
So, in Twin Mode, I can't see any easy method to keep the radio on if there is not an actual connection to the remote RRC box, and if you program them to use only a CAT line for the remote control, most of the comfort disappears that the K3/0 provides.