For my reference and probably others too in the future, here are the configurations that I have to make this thing working:
Microkeyer II configuration:
JT65-HF HB9HQX configuration:
And on my TS-590SG:
Menu 67 - 38400
Menu 69 - ACC2
Menu 70 - FRONT
Menu 73 - 4
Menu 74 - 4