I'm looking for an open problem in analysis or number theory with just one "genuine" or "second order" quantifier.

E.g.

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

- "Every set $S$ of natural numbers 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 an arithmetical problem.

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

CLARIFICATION (added 16 August 2021)

I have examples of such sentences now, but they require specialist background to understand (e.g. functional analysis).

What I'm really looking for is an example that is easy to state and  obviously $\Pi^1_1$ for readers without specialist background.  (And  has no known reduction to an arithmetical sentence.)

While I'd prefer a known problem, I'll settle for a contrived sentence that no mathematician would care about.