All the known bases of combinatory logic, such as $\{S,K\}$, or $\{K,W,B,C\}$,
have one or more combinators using 3 variables:
\begin{align*}
S ={} & \lambda x\lambda y\lambda z. x z(y z), \\
B ={} & \lambda x\lambda y\lambda z. x  (y z), \\
C ={} & \lambda x\lambda y\lambda z. x   z y.
\end{align*}
This raises the question whether we can make a basis with functions of only 2 variables. Surely if such a thing were possible, it would be a most interesting result and it would be well-known.
But as far as I can tell, no such thing is known. Thus it seems nobody believes such a thing to be possible.

But has this been proven anywhere?

How do we know that $\{K,W,2,O,T,D\}$ is not a basis, where
\begin{align*}
K ={} & \lambda x\lambda y. x, \\
W ={} & \lambda x\lambda y. x y y, \\
2 ={} & \lambda f\lambda x. f(f x), \\
O ={} & \lambda x\lambda y. y(x y), \\
T ={} & \lambda x\lambda y. y x, \\
D ={} & \lambda x. x x?
\end{align*}