The only pointclasses I know how to prove the prewellordering property for are the usual $\Sigma^1_0$, $\Pi^1_1$, and $\Sigma^1_2$ (and their boldface variants). Do we know that $\Sigma^1_2$ is the limit of what's provable in $\mathsf{ZFC}$? In other words, is $\mathrm{PWO}(\Sigma^1_{n})$ independent for $n>2$? (Especially even $n$)
I know that $\mathrm{L}\models\mathrm{PWO}(\Sigma^1_n)$ for $n\ge 3$ while projective determinacy implies $\mathrm{PWO}(\Sigma^1_{2n})$ and $\neg\mathrm{PWO}(\Sigma^1_{2n+1})$ for $n\in\omega$. So this would show the independence of $\mathrm{PWO}(\Sigma^1_{n})$ for odd $n$, but I'm not sure what we know about larger $n$ in general. I think I've heard about a forcing extension of $\mathrm{L}$ where $\mathrm{PWO}(\Sigma^1_4)$ fails, but my memory might be wrong, and I don't know whether this can be generalized to the other pointclasses.
EDIT:
Some definitions: for a pointclass $\Gamma$, we say $\Gamma$ has the prewellordering property, written $\mathrm{PWO}(\Gamma)$, iff every $X\in\Gamma$ has a $\Gamma$-norm.
A $\Gamma$-norm on a set $X$ is a function $\phi:X\rightarrow\mathrm{Ord}$ such that the two relations
- $x\le_\phi y$ defined by $x\in X\wedge (y\in X\rightarrow \phi(x)\le \phi(y))$; and
- $x<_\phi y$ defined by $x\in X\wedge (y\in X\rightarrow \phi(x)< \phi(y))$
Are both in $\Gamma$. (Note that if $X$ has a $\Gamma$-norm for adequate $\Gamma$, then $X\in\Gamma$, defined by $x\in X\iff x\le_\phi x$.)