I'd love to see evidence of that. I did some testing of the effects of websockets vs polling for iOS and saw negligible effects.
The OS doesn't power down the antenna while there are open network connections. Did your tests wait long enough to turn off the antenna, was the test environment clean (i.e., nothing else using the network on the device, including your tests)?
Were you testing it on WiFi or cellular?
Even if it's only when the app is not backgrounded?