Michael, I can't see your inserted image. It seems to me that you try to ask your Desktop publishers to put the TWB or TWBX into a shared drive outside Tableau before 4 authorized publishers to publish them into production server (if I understood it correctly). If that is the case, you are creating a lot of extra work to your team since the 4 authorized publishers can't tell easily if the workbooks are good or not from shared driver unless they download the workbook from share drive, and upload them to a Tableau server.
It is much better to create a Staging and Production projects (those projects are also called folders) in your Tableau Production server. All 35 Desktop users can publish to Staging projects while only 4 of them can publish to Production project. The 4 authorized publishers should have project leader permission of both Staging and Production projects while remaining Desktop users have site role as Publisher but have publisher role for Staging project only - which means that all 35 Desktop users can publish to Staging project while only 4 authorized publishers can publish to both Staging and Production projects. With this setup, the 4 authorized publishers can review all the workbooks/data sources in Staging projects to make sure everything is Ok: perf is good, data is accurate, etc. Then the 4 authorized publishers can move the selected workbooks from Staging project to Production project (project leader permission allows them to move workbooks or data sources).
When there is a newer version of the published workbook in Production project, Tableau server will not allow you to move a new version workbook with the same name to Production project even you turn on version control which is true in 10.0. You will get the following error "A workbook with the same name already exists in this project".
What you have to do is to download the new version from Staging and re-publish is to Production with the same name, which will create a newer revision.