Are there any examples of context-free grammars for which we simply do not know whether they are ambiguous? By examples I mean an actual specification for the grammar, not some kind of non-constructive existence proof with a counting argument or whatnot. To make things more clear, I should be able to write a parser for this grammar from the specification (this should exclude grammars that are somehow defined via constructions involving other undecidable problems).
I am not a specialist so I hope the previous paragraph makes what I mean clear enough. Otherwise please do let me know if you think I am not being clear enough.
Another point: coming up with a random grammar specification on the spot and claiming its ambiguity is unknown because no one has tried to figure it out does not count. I guess I am really looking for known open problems in this area.