Should the government increase taxes on private equity executives?

No

I don't believe that this method should be used for government schemes at all. Either don't use it at all, cap it or put penalties in place where PEEs who employ unsafe practices or are responsible for accidents/loss of life are punished.

Yes, but ensure it is part of a broader tax reform addressing income inequality and promoting fiscal responsibility.

It goes to the whole reform tax across the spectrum towards capital and companies, and away from income

They should not be allowed to offset gains with losses made on failed businesses and should pay tax on capital gains at the same rate as income.

They should be taxed slightly more than high earners but all tax should be reduced to incentive wealthy people to live in the UK

Yes, but more generally focused on high wealth & income individuals rather than specifically private equity executives