Nice trick! Here are the debug results:

turn chatui.js.aspx-CHATUI_DEBUG off to hide this frame for default. use CTRL+DBLCLICK the MessageList to show or hide the DebugList

TRACE : Connecting..

TRACE : Connect_Callback . READY ,

TRACE : _Sync_Return:NOCONNECTION:conn-_104

TRACE : Connecting..

TRACE : Connect_Callback . IDEXISTS

What is weird is that if you are able to time it right and send something in that second that it says Connected!, you retain the connection and everything starts working. 

Here is a trace where I hit the connect button and then typed something in and hit return while it still said "Connected!". After this, I retained the connection:

TRACE : Connecting..

TRACE : Connect_Callback . IDEXISTS ,

TRACE : Connecting..

TRACE : Connect_Callback . READY ,

TRACE : CTS:USER_MESSAGE;;dd; ;;;0

TRACE : _CTS:USER_TYPING:

TRACE : * * * * * * * * re sync for xh timeout.

Error : http error1 :0:Unknown:

TRACE : STC:PLACE_UPDATED;;_102;lobby-1;Support;Lobby;0;1;0

TRACE : STC:PLACE_AUTOAWAYMINUTE;;30

TRACE : STC:MYINFO_UPDATED;;USER;_105;User:rick;Rick;Rick;;0;0;1;ONLINE;BYCLIENT;;;;0;Normal

TRACE : STC:ITEM_ADDED;;USER;_105;User:rick;Rick;Rick;;0;0;1;ONLINE;BYCLIENT;;;;0;Normal

TRACE : STC:SYS_INFO_MESSAGE;;Welcome to the FacetoFace Health support chat room

TRACE : CTS:USER_MESSAGE;;dd; ;;;0

TRACE : _CTS:USER_TYPING:

TRACE : STC:USER_MESSAGE;;_105;User:rick;Rick;dd; ;;;0;633591347980554800

TRACE : STC:USER_TYPING;;_105