We tried installing Version:06.05.01.05 but it couldn't Revit 2023.
As previously mentioned, you don't have the ability to use a 4D iModel for a trial project. If you have access to SYNCHRO 4D Pro, you can request your account Admin to set up a proper Control project.
Regardless, the iTwin Connector only supports till 2023. So you may still have to rely on secondary formats like IFC.
Adding Arpit's comment,
If you have acces to 4D pro, Below link will explain how to get the Register Project Button. And also explain how to know who is your Account Admin.
Why can't I register a SYNCHRO Project