The root-class residuality of the fundamental groups of certain graphs of groups with central edge subgroups

Consider a class $\mathcal{C}$ of groups which contains at least one nontrivial group and is closed under subgroups, extensions, and direct products of the form $\prod\nolimits_{y \in Y} X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$. Suppose that $G$ is either a tree product of finitely many groups with central edge subgroups or the fundamental group of an arbitrary graph of groups with trivially intersecting central edge subgroups. We establish some sufficient conditions for $G$ to be residually a $\mathcal{C}$-group.