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The technology to allow them to live the kind of life they would want on Mars would also greatly decrease the human impact of the climate crisis applied to earth. A mars colony fundamentally requires the ability to insulate an entire city, to ultimately civilization, from a climate more inhospitable than we could render earth even if we burned every scrap of coal and oil in the ground, and a billionaire would probably want more than just that bare minimum functional civilization to live someplace.
I don’t think they could actually stop such a technology from being deployed on earth in such a scenario. Nor do I think they would really object, it seems doubtful to me that they would all want to go to mars, and for those that don’t, trying to prevent it’s use is counter to their interests.