We consider the output synchronization problem for a heterogeneous network of linear agents. The agents are non-introspective, meaning that they do not have access to their own state or output; the only information available to each agent comes from the network, in the form of a linear combination of its output relative to that of neighboring agents. The basis for our study is a design previously presented by the authors, which was based on an additional assumption requiring the agents to be right-invertible. Here we dispense with the assumption of right-invertibility and instead introduce a new assumption based on solvability of a particular set of regulator equations.