The higher-rank wreath-product non-embeddability conjecture

Let $A$ be a nontrivial finite abelian group, let $\Sigma$ be an alphabet, and let $\llbracket \Sigma^\mathbb{Z}\rrbracket$ denote the topological full group of the full shift. For $d\geq 2$, let $A\wr\mathbb{Z}^d$ be the wreath product of $A$ by $\mathbb{Z}^d$.

Higher-rank wreath-product non-embeddability conjecture. If $d\geq 2$, then

$$A\wr\mathbb{Z}^d\not\leq \llbracket \Sigma^\mathbb{Z}\rrbracket.$$

The preceding theorem rules out move-$A$ithful actions of $\mathbb{Z}^2$, so the paper's construction cannot produce $A\wr\mathbb{Z}^2$ in this way. It does not exclude other constructions, and the conjecture remains open.