In this note we study a family of graphs of groups over arbitrary base graphs where (1) all vertex groups are isomorphic to a fixed countable sofic group Formula: see text, (2) all edge groups Formula: see text are such that the embeddings of Formula: see text into Formula: see text are identical everywhere. We prove soficity for this family of groups under a flexible technical hypothesis for Formula: see text called Formula: see text-sofically-separable. This proves soficity for group doubles Formula: see text, where Formula: see text is an arbitrary separable subgroup and Formula: see text is countable and sofic. This includes arbitrary finite index group doubles of sofic groups among various other examples.
Gao et al. (Thu,) studied this question.