Bike is back!! New fob “installed”, haven’t ridden with it jet (to cold + insurance stop during winter) but tested in the garage: contact ON, key nearby: no issues. Easily 10 min plus without switching off. When I remove the fob outside range, within 1 min she goes to sleep. So… looks sorted. Cost me a little over € 300,- for the fob, but beats stranding after coffee stop or lunch.
