This question only considers federal taxes in the United States. Also, by "capital income" I am including dividends, interest, capital gains, corporate incomes and estates.
The Chamley-Judd result states that under certain assumptions, capital income should be untaxed to maximize economic efficiency. More recently work has called some of those assumptions into question and suggested that a positive capital income tax is optimal, but I believe that there is more or less of a consensus across economists that from a pure economic efficiently standpoint, capital income taxes should be significantly lower than they currently are.
Nevertheless, many economists oppose lowering the capital income tax rate (possibly offset by a uniform increase in the ordinary income rates) on fairness/distributive grounds, as people who earn capital income tend to be wealthier than average, so the capital gains tax is significantly progressive (at least in the short term, when we can neglect changes in economic growth), and so such a change would probably be regressive.
Opponents of capital income taxation respond that any decrease in the overall progressivity of the federal tax system brought about by eliminating capital income taxation can simply be compensated for by making the ordinary income tax rates more progressive. While there would certainly be individual winners and losers under such a change (e.g. people who earn significantly more or less of their income from capital than most people at their same income level), in principle the distribution of effective overall federal tax rates across the entire US population could be left completely unchanged by sufficiently precise tweaks to the ordinary income brackets. Such a change would boost the tax code's efficiency while leaving its progressivity unchanged. (I'm assuming that everyone pays their taxes honestly and neglecting any changes in tax evasion that this change might cause.)
If all capital income taxation were eliminated in the US, how would the ordinary income tax brackets need to change in order to keep the federal tax system's overall progressivity the same?