In combinatory logic, is there a uniform universal generator?

The source page states the original problem together with its recorded qualifications and progress updates as follows.

Recall that $M$ is a universal generator if each combinator $P$ has a superterm $Q$ such that $M \rightarrow^* Q$. Call $M$ a uniform universal generator if there exists a context $C[\cdot]$ such that, for each combinator $P$, we have $M \rightarrow^* C[P]$. Is there a uniform universal generator? (For Combinatory Logic, if we restrict the context $C[\cdot]$ to be of the form $(N \cdot)$, no such term exists [Statman, 1993].)