Languages that we cannot (dis)prove to be Context-Free I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques.


  Is there a recent survey on the subject or an open problem section from a recent conference ?  
  
  Probably there are not many languages which are not known to be CF, so if you know one you can also post it as an answer.


The examples I found are:


the well known language of Primitive words $Q = \{ w \mid w \neq u^i (|u| &gt; 1) \}$ (there's a whole nice recent book on it: Context-Free Languages and Primitive Words)
the Base-k representations of the co-domain of a polynomial (see question "Base-k representations of the co-domain of a polynomial - is it context-free?" on cstheory, which perhaps has been solved by domotorp, see his preprint)


Note: as showed by Aryeh in his answer you can build a whole class of such languages if you "link" a language to an unknown conjecture about the (non)finiteness or (non)emptiness of some sets (e.g. $L_{Goldbach} = \{ 1^{2n} \mid 2n$  cannot be expressed as a sum of two primes$\}$). I'm not quite interested in such examples.
Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${\bf t} = 0110100110010110 \cdots $.  To give some context, Jean Berstel proved that the complement of the set $T$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that).  But the corresponding result for subwords is still open.