Time and space are first and foremost concepts (ideas, opinions, "facts")
Have you tried asking a mountain whether it ascribes the state "exist" to the concepts of time and space?
It is true that time and space exists for humans thinking about the mountain however.
Time exists for the mountain. The mountain moves through time and is effected by time. The mountain is incapable of perceiving time. Neither can bacteria perceive dimensions, but they are still controlled by them.
A mountain is a part of spacetime. But it does not "exist" for the mountain.
Certainly the mountain is affected by the environment and can be modelled with the spacetime concepts. But the mountain does not perceive anything, and therefore cannot compute isExist (spacetime)