The way the BLS uses housing and health insurance in the CPI is done with a lag and only at certain times during the year. If they have not updated the numbers in recent months but, in fact, they might have been falling or stabilizing, they won't affect the CPI (at least not yet). Then the CPI might over state the true rate of inflation. 
It is possible health and housing have been falling or not rising much, so in the near future, when they get updated again, they will both help keep the CPI down. Right now the CPI might be missing some moderation in prices.
Housing Costs, Inflation’s Biggest Component, Are Poised to Ease: New rents are already softening, but that might not show up in official inflation data for months because of measurement delays by Gwynn Guilford of The WSJ. Excerpts: 
"Housing is influential because it is the largest component of the CPI: 
Tenants’ rent made up 7.4% of the CPI in September, and owners’ 
equivalent rent (OER), which measures homeowners’ costs, made up 24%. 
Shelter’s share of the core CPI, which strips out volatile energy and 
food prices, was an even larger 41.7%. As core inflation rose from 4.6% 
in October 2021 to 6.3% in October 2022, shelter inflation contributed 
around 1.4 points of the acceleration. 
Shelter inflation tends to be “sticky,” meaning that once moving in any 
direction, it is slow to change. This has much to do with how it is 
calculated by the Labor Department’s Bureau of Labor Statistics. OER 
isn’t based on home prices or mortgage payments because home purchases 
are an investment, not just a commodity. Instead, the Bureau of Labor 
Statistics bases OER on what an owner would have to pay to rent her own 
home, drawn from rents in high-homeownership areas. 
Meanwhile, most tenants’ rents change just once a year, and therefore 
don’t respond immediately when rents on new units increase. To capture 
what the typical tenant experiences, the CPI measure of tenant rent 
includes new and existing leases, and thus will tend to lag behind 
measures of new leases only.   
“Because 
they’re looking at the overall pool of rents, and not just new rents, 
it’s like an oil tanker turning—it just takes time for market dynamics 
to feed through,” said James Knightley, chief international economist at ING.
The
 way the Bureau of Labor Statistics calculates the shelter index 
contributes to the lag, because it surveys each cohort of properties at 
six-month intervals, then calculates the price change using a six-month 
moving average. A jump in rent, therefore, takes nearly a year to show 
  up, said Jake Oubina, senior economist at Piper Sandler."
 See also Health-Insurance Inflation Is Poised to Drop Sharply: The subindex of the consumer-price index is about to turn from a driver of inflation into a deflationary drag by Gwynn Guilford of The WSJ. Excerpts: 
"Omair 
Sharif, founder of Inflation Insights LLC, estimated the 
health-insurance index will decline 38% by September of next year from 
this past September. That would mean health insurance would go from 
adding around 0.38 percentage point to the 12-month increase in core 
inflation as of last month, to subtracting about 0.42 percentage point 
by next September, Mr. Sharif said. 
The
 change to the one-month rate that shows up in October will continue at a
 fairly consistent pace for the next 11 months. “When we establish that 
new trend, this trend is likely to persist for a full year,” said Ryan 
Wang, U.S. economist at HSBC. “That’s one of the aspects of this methodological quirk.”
Omair 
Sharif, founder of Inflation Insights LLC, estimated the 
health-insurance index will decline 38% by September of next year from 
this past September. That would mean health insurance would go from 
adding around 0.38 percentage point to the 12-month increase in core 
inflation as of last month, to subtracting about 0.42 percentage point 
by next September, Mr. Sharif said. 
The
 change to the one-month rate that shows up in October will continue at a
 fairly consistent pace for the next 11 months. “When we establish that 
new trend, this trend is likely to persist for a full year,” said Ryan 
Wang, U.S. economist at HSBC. “That’s one of the aspects of this methodological quirk.”
The Labor Department bases the price of health insurance in large 
part on health-insurer profits, which are reported with a lag of about 
10 months. Thus, data in the October 2022 CPI reflect what happened in 
2021.  
This lag is currently amplifying big swings in medical-care spending. In 2020, lockdowns, limited healthcare capacity and consumer reluctance to seek care
 translated to lower healthcare spending and thus reduced benefits paid 
by insurers and commensurately increased net premium income. That showed
 up as skyrocketing health-insurance prices starting in October 2021, 
when the 2020 data were incorporated.
This
 year’s updated data are based on 2021, when consumers caught up on 
preventive care and elective procedures, eating into insurers’ premium 
income, which should translate to a drop in health-insurance prices in 
the CPI."
"the methodological quirk only affects the CPI. While that is closely 
watched by investors, the media and consumers, the Federal Reserve 
officially bases its 2% inflation target on the Commerce Department’s 
separate personal-consumption expenditure price index."
"health services have a much bigger weight in the PCE price index than in CPI."
"The lagged nature of the CPI’s health-insurance index means that it may 
not reflect current underlying price pressures in medical-care services.
 Those are likely building, thanks largely to labor costs. 
Medicare’s 
recently negotiated 4.3% reimbursement rate for inpatient hospital 
services for fiscal 2023 was the biggest increase in around 15 years. A 
large share of how Medicare calculates those rates is based on 
employment cost index data for hospital wages" 
"That measure rose at an annual rate of 5.6% in the second quarter, up 
from 2.3% in the last quarter of 2020. The increase is likely to spill 
over into negotiations now under way between private insurers and 
hospital groups"
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