I don't think it's possible to keep the connection open while the browser is in the background, or when the screen is locked, the reason being that the app is essentially frozen in memory. Here's a quote from a similar question:
the reason you cant keep a network socket open, is that without your app jumping to the foreground when it receives a connection, it cannot respond to network traffic(because if it is not in the foreground its memory content is frozen).
However, I did find this page on Push Notifications for Websites that shows you 'how to sign up your users to receive notifications even when your site is not running in Safari'.
There are some other options: if you want to send continuous updates, you could write an app and either follow the instructions on Apple's site to keep a socket open permanently, or you could configure the app to implement Push Notifications.
I'm sorry I couldn't find a quick fix, but I hope at least one of these options works for you!