In the λ-calculus, which sets have the form {M | Q →* PM}?

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

It has been proved that (in $\lambda$-calculus or Combinatory Logic) every recursively enumerable set of ground terms that is closed under conversion has the form $\{M | PM \leftrightarrow^* Q\}$ for some $P$ and $Q$. Which sets have the form $\{M | Q \rightarrow^* PM\}$?