This is inspired by this question. Let $\mathcal{C}$ be the collection of all combinators which only have two bound variables. Is $\mathcal{C}$ combinatorially complete?

I believe the answer is negative, however I was not able to find a reference for this. I would also be interested in references for proofs of combinatorial incompleteness of sets of combinators (I can see why the set $\mathcal{D}$ consisting of combinators with only one bound variable is incomplete, so these sets ought to contain more than just elements of $\mathcal{D}$).