# Presburger arithmetic: is it known to be in $EXPSPACE \setminus EXP$?

My understanding of the Presburger arithmetic decision problem is that it requires doubly-exponential time, but only singly-exponential space, meaning that it is in $EXPSPACE \setminus EXP$. Obviously, if this set is not empty, then $P \subsetneq PSPACE$, solving a major open problem, so my understanding is probably wrong. Where did my reasoning fail?