Is he double-counting? I honestly don't know.
If each of his indexes contained the same info, the average would be correct. The problem is using housing prices. I fought over this with that goof Paul Ross. If housing prices increased by 12% but only 1% of the population bought a house then 1% x 12% = 0.12% increase in the basket of consumer goods used to calculate CPI. More people buy gasoline, so an increase hits a larger % of the population, but the total spent per year is much smaller compared to a house purchase.
CPI-U is supposed to be based on what a "typical" urban dweller consumes in a year. That's where Paul Ross was wrong. He had a source that showed people buy a new house every 12 years. I "estimated" that that meant only 8% buy a house in a given year. So, 92% who don't buy are typical, 8% who do are atypical. Conrad treats the housing increase as hitting 100% of the population. Just a small error on his part. LOL!!