If oil is constant as you assert, this is severe deflation, not inflation.
You cannot have it both ways.
The dominant economic trend is deflationary. I’ve never said otherwise.
Beyond that you can have price inflation/deflation every which way. Oil goes up, real estate goes down. Note that for the exurbs, these two phenomena are actually related.
Again, being an illiquid asset, real estate is subject to different price controls than a commodity like oil. A barrel of oil costs the same whether I’m buying it in San Diego or Detroit. The same can’t be said for housing.