Inflation is a separate problem.
Back in the real world however, the relative price/value/production of things is not a separate problem...and since I'm guessing most people's objections to inequality is not its effects in some hypothetical world where all concepts are independent, but in the real world where they all connected, I'm not sure hypotheticals of this nature are often that useful...
If you want to claim that inequality somehow causes poverty (and poverty is the real problem), make that case.
> Inflation is a separate problem
It's not just inflation: supply/demand dynamics will be changed across income levels. For example bidding wars might cause rentals to go up, and the mother will be outbid (see: gentrification or recent news on San Franciso rents)