Using PS Query based Pop Select - What is minimum security to allow the GetQuery method to work

We are attempting to provide users the minimum level of security to execute a PS Query-based PopSelect associated with a process. We have added the Query Access Manager security to the records used by the Queries associated with the process. The list of queries shows fine, but when the user selects the query, we get an error indicating the Query cannot be opened. Based on the trace, this error is coming from the Open method associated with the Session.GetQuery api.

We really don't want to provide access for these users to Query Manager or Query Viewer. What security is required to be able to use the Open method of GetQuery?