 Solution

A closed beta-eta-long normal term of type $\sigma \to \sigma \to \sigma$ must have the form $\lambda x:\sigma .\lambda y:\sigma .t$, where the normal term $t:\sigma$ can only be $x$ or $y$: the pure calculus has no constants or other closed source of a value of the atomic type $\sigma$. Hence the only two beta-eta-equivalence classes are the Church Booleans