To the extent that purchase prices vary independently of rental prices in the same market, the CPI inclusion of equivalent rents will not reflect home purchase prices.

Then again, I don’t see why it should, unless it is some measure based on the lower of rents (actual or equivalent) or amortized purchase and ownership costs (as you need shelter one way or the other), but that could only drive the CPI down compared to just considering rents as is done now.