I'm looking for an open problem in analysis with just one "genuine" quantifier.

E.g.

• "Every continuous function $\mathbb{R} \rightarrow \mathbb{R}$ has the property $\theta$", where $\theta$ is expressible using only quantifiers over rationals.

No cheat examples like "For every real number, Goldbach's conjecture holds"! That's a problem in number theory, not a problem in analysis.

In technical terms, I'm looking for a $\Pi^1_1$ sentence that we don't know how to reduce to an arithmetical sentence.