Imagine people become more egalitarian, to the point where they heap scorn on the rich and successful. What is the effect on inequality? By the previous logic, the effect is directly counter-productive. The more you scorn rich people, the more people you scare away from high-income professions. The more you scare away, the lower their supply. And the lower their supply, the higher their income!
Lesson: If you really want a materially more equal society, stop beating up on the 1%. Do a complete 180. Smile upon them. Admire them. Praise them. Sing songs about how much good they do for the world. The direct result will be to raise their status. But the indirect result will be to pique the envy of status-conscious people, increasing the competition among the top 1%, and thereby moderating income inequality.
On the other hand, if you want to increase material inequality, by all means heap scorn on the rich and successful. Try to fill them with guilt and self-loathing. The 1% who remain will find that living well is the best salve for their consciences.