The issue is that in the process of authenticating to the network, it goes through at least three iterations...
- First the phone tries to connect via the minimal cellular connection (X1)
- Then the phone attempts to connect to 3G and authenticate for data
- Finally it attempts to authenticate for 4G Data
Those processes are far more taxing on the phone's battery than the simple pinging it does to a network that it's already authenticated on. Now, if your 4G is strong in your area it won't tax all that much more than the 3G, but if the range is farther away it will eat up a lot even during the simple pinging, so in that case, I'd shut down 4G, especially since you are in a WIFI network.