Is there a one-rule string rewriting system that is non-terminating but also non-looping?

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

Is there a one-rule string rewriting system that is non-terminating but also non-looping, that is not allowing a rewrite sequence $x \rightarrow^* uxv$ ? Motivation and examples of two rule string rewriting systems and a one rule term rewriting system that is non-terminating and non-looping, can be found in [Zantema, 1999].

This problem is related to RTALooP entry one and RTALooP entry string-rewriting. More about the history of this problem in the context of the question of one-rule termination can be found in [Zantema, 1999].