Does $\exists\mathbb{R}=\mathbf{PSPACE}$? The complexity class $\exists \mathbb{R}$ (the existential theory of the reals, i.e. problems that can be reduced to deciding if a collection of polynomial inequalities has a solution) is known to be contained in PSPACE and to contain NP. See this paper for a reference; as far as I can tell, that's the paper that introduced $\exists\mathbb{R}$ as a distinct complexity class.

However, Reif gives an explicit construction which takes a Turing machine with a polynomial space bound and a string, and constructs a motion planning instance which has a solution if and only if the machine accepts the string. Since motion plan existence can be reduced to $\exists\mathbb{R}$ (see, e.g., this paper by Schwartz and Sharir), this seems to imply that $\mathbf{PSPACE} \subseteq \exists\mathbb{R}$, and thus that $\mathbf{PSPACE} = \exists\mathbb{R}$.

I don't come from a CS theory background, and I feel like I must be missing something fundamental, because there looks to be a whole subfield  of authors writing papers placing various problems in $\exists\mathbb{R}$.
The issue may be whether or not Schwartz and Sharir show that motion plan existence is many-one polynomial time reducible to $\exists\mathbb R$.

If they need several queries to $\exists\mathbb R$ for a given motion plan existence instance, then that's not a many-one reduction.