Ok, I just tried again.
Turned off WiFi and put phone into Airplane Mode. Verified that Bluetooth is still connected to the watch.
Go into the Connect app. There's a big red box that says "No Internet Connection."
Heart Rate is transmitting to the Connect App in real time.
However, if I try to run a Sync from Connect, it hangs and then fails.
Turn on WiFi and take phone out of Airplane Mode. Run a Sync from Connect. It completes successfully.
Confirmed - If you complete an Activity and want to sync your watch to your phone with the Connect app, your phone must have an Internet connection.