Of course I don't know the internals of your code, but from playing around with it, it seems to me that it wouldn't be that hard to fix.
After it says "Connected!" and before it says "Connection to the server is lost", if you manage to type in a message and send it during that window. It seems to retain the connection and work fine after that. Perhaps you need to add something to your application to mimic this behavior.
We have a quad-processor server, which is pretty common these days. If we don't use a web garden, then we are not taking full advantage of the multiple processors.
Do you have any plans to address this issue, or are you going to limit yourselves to the single-processor server market?