Basicly the problem is caused by this:
When judged the saints get aionios life and the wicked aionios torment.
The duraration of what they get is the same.
Elsewhere God is called aionios God. He existed forever so the ET argument is that the duration of the word is forever in all 3 cases.
Those who believe the wicked are anilihated argue the effect of the punishment is eternal. God is eternal. Life is eternal.
UR is a little more complex.
Some believe the word never means eternal. There are ages before and after.
Some believe the meaning of the word depends on the context in which it is read.
The purpose of the article on the site is to prove punishment (=ET hell) is not eternal.
So it's not about His Spirit or anyone elses spirit dying; but how long the suffering for that eternal spirit lasts.