We're talking about the poor here, not the middle class. A lot of states haven't raised their minimum wage above the federal minimum wage, which hasn't been raised since 2009.
Also even if wages go up inflation still reduces the wealth owned by the middle class. Especially given that interest rates on savings hasn't kept pace with inflation for decades. So unless you own a house or a stock portfolio even the middle class gets poorer due to inflation.
Sure that's an if, but the effects of inflation certainly exacerbates that inequality. Thus the rich get richer while the poor get poorer. It's just a question of cause and effect.
If the rich getting richer drives up inflation then that does indeed mean that the poor get poorer.