Here are two *non*-examples, one erring in each direction:

 - **Too simple**: "The continuum is an $\aleph$-fixed point," that is $\mathfrak{c}=\aleph_\mathfrak{c}$. This is equivalent of course to $\mathfrak{c}\ge\aleph_\mathfrak{c}$, which means it can be expressed as "There is an $\mathbb{R}$-indexed family of sets of reals of pairwise distinct cardinalities," which is $\Sigma^2_2$. Contra my original guess, however, this *does* have a $\Pi^2_2$ equivalent observed by Farmer S in the comments below *(and I'll add his argument here later when I have more time)*.

 - **Too complicated (so far!)**: "The continuum is a limit cardinal." This can be expressed in a $\Pi^2_3$ way *(which I originally miscounted - thanks to Andreas Blass for bringing this to my attention)* as "For every set of reals $X$, either there is a surjection $X\rightarrow\mathbb{R}$ or there is a set of reals $Y$ such that there is no surjection $X\rightarrow Y$ or $Y\rightarrow\mathbb{R}$," and I don't see how to get a $\Sigma^2_3$ equivalent even granting large cardinals. *(In particular, note that "The continuum is $\ge$ some uncountable limit cardinal" is easy to express in a $\Sigma^2_2$ way as "There exists an $\omega$-sequence of sets of reals of strictly increasing cardinality," but this doesn't seem to be useful here.)*

Of course, the first example doesn't work, and the second example almost certainly doesn't work. That said, I think this is a good indication that general continuum combinatorics is a good place to look for high-complexity third-order sentences.