I haven’t seen enough data to support websockets being a poor fit for mobile, at least in the context of day-to-day average use.
An Energy Efficiency Study of Web-Based Communication in Android Phones (hindawi.com)
There were numerous complaints about battery drain on some android apps that tried to implement chat using websockets. It mostly had to do with the antenna maintaining a persistent connection. The ones that used MQTT performed better and those using push notifications performed better. I dont have the numbers or the implementation details though.
