This question is about logical complexity of sentences in third order arithmetic.  See [Wikipedia](https://en.wikipedia.org/wiki/Arithmetical_hierarchy) for the basic concepts.

Recall that the Continuum Hypothesis is a $\Sigma^2_1$ sentence.  Furthermore (loosely speaking) it can't be reduced to a $\Pi^2_1$ sentence, as stated at [Emil Jeřábek's answer to _Can we find CH in the analytical hierarchy?_](https://mathoverflow.net/a/218649/170446).

Is there an example of a $\Sigma^2_2$ sentence with no known reduction to a $\Pi^2_2$ sentence?  (Equivalently, a $\Pi^2_2$ sentence with no known reduction to a $\Sigma^2_2$ sentence.)  I mean that there should be no known reduction even under large cardinal assumptions.

I'd prefer an example that's either famous or easy to state.  But to begin, any example will do.

*Update:* Sentences such as "$\mathfrak{c} \leqslant \aleph_2$" and "$\mathfrak{c}$ is a successor cardinal" are $\Delta^2_2$, meaning that they're simultaneously $\Sigma^2_2$ and $\Pi^2_2$. The reason (in outline) is that each of these sentences can be expressed "$\mathbb{R}$ has a well-ordering such that..." and so can its negation.