first of all, my problem solved, with a new lighting cable from AirPods 2 box, 11 pro max works flawlessly with CarPlay now.
Please see below are my answer:
1) 11 pro max running 13.2 and 13.2.2 and 13.3, all of them are not working with CarPlay with old lighting cable
2) I just have my 11 pro max for 1 week, before this, I have my XS MAX, and works flawless with CarPlay
3) my Apple CarPlay is after market, never been upgraded. and to my best knowledge, Apple CarPlay device is just mirroring the iOS system, like a TV receive signal through HDMI cable from a XBOX. Therefore, there’s no need to upgrade your CarPlay device
just go buy a new lighting cable, try it, then let me know how this works for you.
Thanks!