The notion that changes in interest rates, which generally correlate with inflation, would alter the price of stocks is not obvious to me. Not sure what mechanism you would propose that would cause that to happen.
Higher rates -> reduced accessibility of personal credit -> lower spending -> lower corporate revenue -> higher cost of borrowing -> higher cost of debt service -> lower profits -> stock price
Rates act as a global parameter that impacts performance of companies differently, with the consensus understanding that it impacts debt-fuelled (or what we now usually call growth) companies the most.