A cardinal $\kappa$ is $\Sigma_n$-correct iff $V_\kappa \prec_n V$. For n>1, how to force a $\Sigma_{n+1}$-correct cardinal to be $\Sigma_{n}$-correct but not $\Sigma_{n+1}$-correct?

For $n=1$, we can force GCH below $\kappa$ and then violate GCH at $\kappa$.

A cardinal $\kappa$ is $\Sigma_n$-correct iff $V_\kappa \prec_n V$. For n>1, how to force a $\Sigma_{n+1}$-correct cardinal to be $\Sigma_{n}$-correct but not $\Sigma_{n+1}$-correct?

For $n=1$, we can force GCH below $\kappa$ and then violate GCH at $\kappa$.

If we assume some large cardinals, there are more partial answers, but the general situation is not clear to me.