The built-in state persistence saves all properties it can. An idea for granular and/or configurable set of properties for saving already exists in our Ideas and Feedback portal and you can cast your vote and follow its status here: https://feedback.telerik.com/Project/108/Feedback/Details/135962-extend-the-radparsistanceframework-to-be-able-to-set-specific-rangfe-of-settings. It would be helpful if you add a description of how you would expect such a feature to work and be exposed for configuration. If it gains traction with the community, our management will consider its implementation.
If you want to save only certain properties, you will need to implement a custom persistence and get/set the desired properties yourself, which can become a little tricky: https://docs.telerik.com/devtools/aspnet-ajax/controls/persistenceframework/functionality/store-custom-settings.