Went back to my home for a few weeks, my Airport express kept all my connections alive. Came back to Tokyo, tried my SMC router, all my iDevices disconnected from wifi when asleep.
Set up my Airport Express in my Tokyo office and all my iDevices keep a permanent wifi connection, asleep or not, now I get all notifications.
I retried the old SMC router because I had installed iOS 7 on my iDevices, I thought the new OS may have cured the problem, it made no difference, they still disconnected on my SMC router.
So for me, the solution has been, buy an Apple router.
I came to this solution after going to The Apple store complaining about my wifi connection on iOS 6, only to find that it never disconnected in the Apple Store, went to a friends place, he has Apple routers, it never disconnected there either, came back to my place, disconnections on SMC router, so the SMC router was the obvious problem.
For me and like many others, these Problems started on iOS 6, and changing routers has been the only solution that has worked for me, some other routers work fine, I had an old DLink router that kept connections alive too, no matter what the iOS, but it was an old and now discontinued model, and the newer model from the same manufacturer did not keep the connection alive. 😮
I must have tried every solution on the internet, but it was proven to be a router problem in my case, some routers will keep my connection alive, and some will not.
Your Mileage May Vary.