CPI works by asking how much people pay for rent. If home prices raise 20% in one year (not at all unreasonable in various times in the last ten years), it takes a long time for that to be reflected as many people have their rents fixed, some people have rent control, some landlords will only raise rents on new tenants, etc.
for existing, well established renters, sure. For new renters, or people who move, or people who face housing insecurity, or people who want to buy homes, it's much higher and not accurately reflected in CPI.
What do you find controversial, and would cause a material difference in the headline inflation rate?