This request is from Zuora.
Our product team confirmed that the stage does not get automatically refreshed for the archived projects in Responsive in real time or in any interval based on the changes made in Salesforce. Any updates made in Salesforce will be reflected in Responsive as soon as a user accesses the project in Responsive.
The client would like to have projects synced with updates, without having to enter the projects everytime.
CC: @Sathishkumar GK