You could try using PAGE context, perhaps.
To clarify, @Begin a conversation in a PAGE scoped component during the GET? Wouldn't the PAGE scoped component get destroyed at the end of the lifecycle making it unavailable if the conversation is then carried out?
How are other folks handling this? This issue would come into play with any global navigation combined with conversation-on-GET, for example. Is it just becoming the best practice to let these conversations expire out?
You can manually pass state from PAGE context to CONVERSATION context. Look at how the booking demo passes SESSIOn scope state into the CONVERSATION context.