The decidability equivalence for Schreier graphs of contracting self-similar groups

Let $G$ be a contracting self-similar group. Its limit space is the associated limit space of the self-similar action, and a limit space is finitely ramified when it has finite ramification in the sense used for self-similar groups. The domino problem and the monadic second-order theory are considered on all Schreier graphs of $G$.

Decidability equivalence. The following are equivalent:

1. The domino problem is decidable on all Schreier graphs of $G$.
2. The monadic second-order theory is decidable on all Schreier graphs of $G$.
3. The limit space of $G$ is finitely ramified.

This conjecture proposes a precise equivalence between algorithmic tiling decidability, monadic second-order decidability, and finite ramification of the limit space for contracting self-similar groups. The supplied text gives no resolution, so the conjecture remains open.