Uniform Universal Generators

Call a closed term $M$ a uniform universal generator if there is one fixed nontrivial one-hole context $C[\,\cdot\,]$ such that, for every closed term $N$,

$$M \twoheadrightarrow C[N].$$

Does such an $M$ exist (i) in combinatory logic with weak reduction, or (ii) in lambda calculus with $\beta$-reduction or $\beta\eta$-reduction?