Are there hyper-recurrent combinators?

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

A term $M$ in Combinatory Logic or $\lambda$-calculus is recurrent if $N \rightarrow^* M$ whenever $N \leftrightarrow^* M$ (this notion is due to M. Venturini-Zilli.) Let's call $M$ hyper-recurrent if $N$ is recurrent for all $N \leftrightarrow^* M$. (Equivalently, $M$ is hyper-recurrent if $P \rightarrow^* Q \rightarrow^* P$ whenever $P \leftrightarrow^* Q \leftrightarrow^* M$.) Are there any hyper-recurrent combinators? (The problem comes up immediately when the Ershov-Visser theory [Statman, 1993] for $\leftrightarrow^*$ is applied to $\rightarrow^*$. It is known that hyper-recurrent combinators don't exist for Combinatory Logic [Statman, 1993].)