I have been experimenting with cables, and it looks like the iPhone 11 Pro Max needs MORE BANDWIDTH in order to support CarPlay PLUS Apple’s new battery optimization algorithm. The old USB-Type-A-to-Lightning cables don’t have enough bandwidth to transmit all the data you need for CarPlay while also transmitting the power needed by your iPhone 11 Pro Max.
SOLUTION. Use an Apple-Authorized, USB-Type-C-to-Lightning cable, and attach a generic Type-C-to-Type-A adapter. I’ve been using both Apple’s own cable that comes in the box with my 11 Pro Max — and today I also tried the Anker Type-C-to-Lightning Cable. Both successfully boot CarPlay for my 11 Pro Max, 100% of the time,
HYPOTHESIS. I figure these newer, Type-C-to-Lightning cables have a newer chip that transmits more bandwidth. They were originally used to fast-charge and transmit data to the new iPads Pro (the ones with Type C connections), and they must have a wider gate for this task. Thus enables them to transmit all the data needed for CarPlay while also charging the iPhone 11 Type C.
Eventually, Apple can solve this with a software, and I understand iOS 13.2 does just that.