Jabra elite sport as BT HR monitor

Until recently I used a fenix 5 and Jabra elite sports in the following configuration: Headphones coupled to smartphone for music, and simultaneously coupled to fenix 5 as an external bluetooth HR monitor. The elite sport measures HR optically from the ear which is much more accurate than wrist based (especially for activity with short, high HR spikes like in the gym).  

Now, with my fenix 6 pro, I can couple the same headphones to the watch to listen to music, but it seems they don't work as an external HR sensor. I tried removing them as as headphones and only add them as HR sensor,but that didn't work either. 

Ideally the elite sport would work as both headphones and HR monitor when coupled to the watch. But for me it would also be ok to choose. E.g. for steady state cardio (running at a constant speed), I would use them to play music from my watch, rely on the wrist HR, and can leave the smartphone at home. On the other hand, when in the gym, I have my phone with me anyway and would couple the jabra elite sport to the phone as headphones for music and to the watch as a bluetooth HR sensor. 

Has anyone tried this combination? Maybe there's a order of pairing and/or connecting devices (phone, watch) that I'm not aware of? 

I know it's quite a special configuration, but if it works its really practical to get accurate HR readings, and therefore calorie estimations, without having to wear a chest strap. I don't complain about the wrist HR, it's a known limitation of this technology.