Sounds like you're doing everything right - it worked for me first time - and that's saying something :-).
1) Install Jabra Sound+ App (on iPhone, may be called differently on Android).
2) Click on "Headset" in the navi below (after you have paired them via Bluetooth natütterli)
3) Click on "Headset firmware" at the top of the list (here you will also find the link to the manual)
and now it's by memory - mine are updated
4) Update firmware - be patient until the yellow stripe on the far right arrives and the app tells you to put the buds in the case.
5) Purple LED flashes for a good 10 to 15 minutes.
After that, everything is fine.
To be on the safe side, I leave both the phone and the case connected to the power supply - you never know.
Maybe a reset (see manual).
https://www.jabra.com/supportpag...
I did a quick search and found no discussion around difficulties with the firmware update. If it still doesn't work after a good night's sleep and some valerian tea, it may be DOA.
Good luck.