Hello Anon,
Your post really isn't related to the prior questions, which are also quite old, so it's best to start a new thread.
As I undestand it, a "Jr" doesn't become a "II."
If John Paul Jones has a son, Steven Jones, and Steven Jones names HIS son John Paul Jones, then, the grandson is Joahn Paul Jones, II.
Jr. stays Jr all his life, I believe.