How does emotional intelligence connect to the idea of avoiding the “promotion trap”?

Will being an emotionally intelligent leader ensure that you would never be seen as an example of the Peter Principle?
To answer this question, please be sure to include your thoughts on these questions in addition to your own points:
How does the Peter Principle look when it plays out in an organization?
What are its main effects on an organization?
How emotional intelligence connects to being an organizational leader?
How does emotional intelligence connect to the idea of avoiding the “promotion trap”?