As the world transitions to Bitcoin, demand for fiat currency will plummet and it will lose all of it's value much quicker than could realistically be caused by central bank printing. As regular people hold more of their net worth in cash compared to rich people, the collapse in the value of fiat currencies will hurt poor people the most, and will not hurt people with assets at all.
Do you think the government will succumb to pressure and allow bitcoin replace FIAT? Bitcoin is not going to replace FIAT, it will only serve as a complimentary currency that will be used alongside FIAT by specific people or industries for some kinds of transactions and this will be followed by some regulations imposed by the government to ensure that bitcoin doesn't become a "threat" to them. What this means is that, not everyone will adopt Bitcoin, people will still have the option to use FIAT and the government will not stop printing currencies. The government is already aware of what would happen if they allow bitcoin take over. For that reason, they would do everything possible to make sure FIAT stays so revenue can still be accruing to the government. Bitcoin has very little impact on wealth inequality which will neither reduce or make inequality worse.