Is it possible to get an overall picture? Do you need to change the plaster behind the stone, or can it be joined before?
The "fake stone" is essentially a tile. The answer is no, unfortunately, it's rarely a good idea to take these down and put them back up again. The adhesive mortar often adheres very well, so the likelihood is very high that some tiles will break when trying to remove them.
Once they've been taken down, the backs will have a lot of old adhesive mortar that needs to be sanded/chipped away before they can be put up again (almost impossible to make them look neat otherwise), a very time-consuming, dusty, and laborious task.