Earlier quoted context omitted.
I think the reason this hasn't been done is you have no way to decide how many recursions are necessary at train time. And if you pick a random number/try many different levels of recursion, you 'blur' the output. Ie. the output of a layer doesn't know if it should be outputting info important for the final result, or the output that is the best possible input to another round of recursion.
Yes, I think training this model would be hard. Perhaps something akin to how MoEs are trained where you impose some sort of loss distribution to encourage equitable routing, but for recursion.
The default mode network does recursive/looping processing in the absence of external stimuli and world interaction. Multiple separate modules outside of the network are responsible for stopping and regulating this activity.