Honestly, I don't think there should be significant difference between asset price inflation and consumer price inflation, at least not in the long term.
This was basically the point I was making. But over the last two decades plus, rich-world consumer price inflation has been seriously and continuously below asset price inflation. What do you think has been happening? And what will happen when the cumulative effect of the difference must eventually be cleared out? (That is, we have much more asset wealth than the wealth in products and services, at current prices -- these must somehow equalize eventually.)
It may be so that CPI is somewhat lagging behind PPI but on long enough timeframes they should follow each other pretty close. Besides, the inflation reports can be rigged year in and year out, but you simply can't rig them year after year since the cumulative discrepancy would be too obvious for everyone to notice and point a finger at the government.
This seems reasonable, but I'll give you an example. I've calculated that the price of a basic, cheap, clean motel room in the US over the last 20 years has really risen on average 6% per year, compounded. Yet the CPI over this period has been 2% in the worst years. Granted, motel rooms aren't part of the CPI 'basket,' the conclusion I had to draw was that, the public don't really hold the CPI accountable over the long term, since few people carry out and publish calculations like this. On a year to year basis, the CPI may be a little on the low side. Over the decades, the cumulative difference is big.