So to address this question, I did the following experiment:
Turned on heart rate broadcast on my Venu 2 and verified it was connected to my 530, and went for a ride. I arrived at my destination, and put the Garmin into sleep mode, which also turned off my radar. It did nothing to the watch, and I did not touch my watch. The watch continued to broadcast for the next hour and a half. Every so often, it would vibrate to remind me to turn off the heart rate broadcast, but I ignored that. After I finished what I was doing, I biked home. I had to turn the radar on manually when I woke up the Garmin 530. Theheart rate continued to broadcast and automatically rejoined. I did not have to do anything at all.
I assume this is the default setting, because I did not consciously change anything about heart rate broadcast mode (nor am I aware if this is even possible).