From the article, the biggest discrepancy between CPI and true inflation is a result of a change in the formula made in 1987 that removed interst rates aka the cost of money from CPI. The CPI formula replaced the actual cost of financing homes and autos with a new term, owner's equivalent rent. The argument made by the underlying paper [1] is that the older calculation better reflects the cost of living increases experienced by consumers and more closely correlates with the widely used University of Michigan Index of Consumer Sentiment. Also, notable is that consumers care much more about the cumulative multi-year effect of inflation while economists only worry about year/year changes in the rate of inflation as can be seen here [2] (worth a click)
The really interesting thing about their proposed measure is that it reduces the period of extremely low interest rates post GFC. We'd be living in a very different world had this been our north star throughout that decade.
[1] https://www.bls.gov/pir/journal/gj02.pdf
[2] https://twitter.com/darioperkins/status/1770783161330323925/...