Should Messages go to Apple Watch when connected to cellular?

I have an Apple Watch Series 4 (connected to Verizon Cellular).

When I go out for a run/walk/bike ride I leave the iPhone XS home and just take the Watch. It does connect to Verizon Cellular when out of Wifi/BlueTooth range of the iPhone, but Messages don't come through to the Watch. I was under the impression that if the iPhone is either Locked or Turned Off the messages are supposed to reach the Watch. I have spoken to both Verizon and Apple but it seems like it is still happening. Any ideas if I am interpreting the "rules" incorrectly? Settings for Notifications on the Watch are set (at least for Messages and Phone) to Mirror the iPhone.

Thanks.