Income inequality is a problem of massively unequal political and economic power, and of insider knowledge and class-cultural arbitrage - not a problem of income.
If - hypothetically - all the jobs required to provide food, shelter, and other basics were taken over by robots, there would be no reason at all not to provide them for free, and give everyone the political and economic freedom to do whatever they wanted.
It's a fair bet that while most people would do nothing much of interest - including most of those who are currently rent seekers - there would still be enough competitive drive to inspire a decent chunk of the population to excel in art, music, science, math, and other productive activities.
That would be real freedom as a birthright - not as the freedom from economic slavery that can only be earned by contributing to that slavery, as now, but as a true inalienable right.