I do not see why this has to be true. Change in prices (inflation) is very different for different people or groups of people, because the goods and services they buy are very different.
At the same time though long-term commodities inflation is cyclical, while wage inflation tends to be sticky. If wage inflation actually causes raises in wages for working class people and isn't just contained to SWEs then long-term this is likely beneficial and would lower disparity in the long run. But this would take a decade of wage-price inflation. Short term it is a regressive tax.
But the Fed is trying to go back to the conditions of the past 40 years where wage inflation has been capped at around 2% which will just squeeze the lower classes even more.
Why would this always have to be true? For example, if the supply of labor sellers at the bottom decreases due to decreasing birthrates, immigration, or people who are no longer employable due to drug addiction. It seems conceivable to me that the lowest paying jobs would have to contend with decreasing supply of labor by increasing the pay.
Perhaps the increased pay then causes increased prices for goods/services which then cause sufficient decrease in aggregate demand that demand for the labor then starts decreasing so it pushes wages at the bottom back down , but I do no see why that has to be case.
I mean, this is a question of policy. The US Federal Government mandates that crops be burned as fuel, which hugely pushes up the price of corn and in turn that pushes up the price of other things that could be grown using that land. We don't explicitly buy up and destroy crops anymore, or pay farmers not to grow them, but the food supply is being deliberately contracted as a matter of federal policy to keep farming profits high.
It seems bonkers to say that there's nothing we could do about rising food prices when we're literally paying to (from the perspective of the food supply) destroy food and keep it scarce.