  In this paper, we study the descriptive set theoretic complexity of the
equivalence relation of conjugacy of Toeplitz subshifts of a residually finite
group $G$. On the one hand, we show that if $G = \mathbb{Z}$, then topological
conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast,
if $G$ is non-amenable, then conjugacy of Toeplitz $G$-subshifts is a
non-amenable equivalence relation. The results were motivated by a general
question, asked by Gao, Jackson and Seward, about the complexity of conjugacy
for minimal, free subshifts of countable groups.
