I'm unaware of any team that has laid "claim" to private property. Other teams have been granted access to Rose (there are 3 permits) as well as Wakulla (deep cave team and WKPP). I think you're mistaking research groups with groups designed to allow access to cave systems.
I've participated in 3 different Florida based permits, by simply submitting my diving resume and offering to help with the organizations goals while following their procedures.
Try to keep up with me here. There ARE caves outside of Florida. I said PUBLIC not private.