Just some further "homework" I've done following Alan's wiring mentioned above - on my Kluger Hybrid 2022 GXL (LED headlamps) I found the LH headlamp assembly had the wires this colour, the RH assembly were different colours but unable to disconnect the plug due to A/C pipes cramping my access. Testing the voltages as Alan posted, the Purple wire provided 12V when activating high beam by pulling the stalk forward ("flashing" the lights) but with engine on, there was 12V there whether high beam was selected or not, so it looks like the Purple wire supplies 12V for the lighting assembly as a whole and is not differentiating between high or low beam (due to being negatively switched).
I tested a number of other wires and couldn't get any significant readings, so I tested the Yellow... bearing in mind that the Kluger Hybrid (like many others) is negatively switched, this seems to be the correct wire for activating high beam. With engine off, there was uncertain voltage that dropped to clear '0' when flashing, with engine on, this line st at ~5V and dropped to '0'V ONLY when the high beam was selected by either pushing or pulling the stalk.
So would I guess the purple would be the 12v source if you were wiring the driving light relay coil across the purple-yellow to provide the trigger for the additional high beam relay (I assume that is what Alan has done?).
I'm also wondering whether the 5~6 volts sitting on that line when the high beam is off would cause the relay to 'stick' on and not turn off when the high beam was dipped - it might only happen randomly.... Hopefully Alan can reply and provide a little more detail -for example, I noticed on Narva's website for wiring negative switched lights that they add a diode in line with the relay coil to ensure current only flows in one direction?