A lot of investigation later:
Seems this time I have a really big problem with this. I have a new component which is very similar to the ones I already had (did not find any convincing difference until now), but it does not work.
The new component has a @Create @Begin method to begin the conversation, just like all of my other components.
It is @Conversational, just like all of my other components.
And it is called from rich:dropdownmenu with conversationPropagation=none or end, I tried both, just like all of my other components.
But I get the annoying "Begin method invoked from a long-running conversation" error.
Now I tried to use join=true (not that I wanted to do this, but only to look what will happen), and the result is very impressive:
"no long-running conversation for @Conversational bean"
I've tried to debug all this. Set a breakpoint at Manager.setLongRunningConversation. But this was a dead end. Conversations are turned on-off-on-off just during one single Event (and sometimes just on-off-on, that's the problem), which gave me absolutely no clue.
Please help me!!!
Finally I've found the cause.
The page which does not work has page parameters in pages.xml. If I remove them, that page can be called (but I need them).
But now that I found the reason, I still don't know why this is an error and what I can do.
Can you help?