I have 2 EEs and I've seen both in the nests so I know both are laying. I have 2 Australorps and 2 Delawares. They all began laying about 3 weeks ago. An EE egg was the 3rd one I got.
Now my 6 birds are giving me consistently 5 eggs a day... only one EE a day ever.
The other 2 breeds are all laying daily, but my EEs seem to be taking turns.