Not sure where you live but cold affects the Auto Stop. In chicago mine basiacallly didnt activate for a few weeks i had no idea what was going on. It got warmer and all the sudden it worked. I called LR said if its below like 30 to expect it to not engage.What is happening to me is that 8 out of 10 times the Eco Auto Stop does not work. When it does engage, the RPM shuts down for a bit but them kicks back in even with my foot still on the brake at a red light. LR said they found nothing wrong with it but its still happening. It's back in the shop on the issues now so hopefully they will look at that also.
Also after they washed it last time got condensation in the tail lights lens....addressing that issue also
I have never had the engine shut off completely.