Soficity preservation under the induction operation

Let $A$, $G$, $\Gamma$, and $J$ satisfy the hypotheses of Lemma~, and let $\iota_J:A^G\to A^\Gamma$ denote the induction operation. A subshift $X\subseteq A^G$ is sofic if it is the image of a shift of finite type under a uniformly local definable map. Soficity induction conjecture. If $X\subseteq A^G$ is a sofic shift, then $\iota_J(X)$ is a sofic shift.

The preceding theorem proves this when the induction of every shift of finite type is again a shift of finite type. The conjecture asserts that this extra hypothesis is redundant.