You asked Mark Carney what the best indicators were for inflation. Let me take the liberty of giving you mine.
1) Median CPI inflation, i.e. the weighted median value of CPI inflation across products. This measure tracks the underlying signal in inflation because it filters out volatile shocks hitting certain industries (e.g. airlines or used cars now, healthcare during 2010-2015, food and energy perennially). Median CPI has a good time series correlation with unemployment, better than the other series (see Ball & Mazumder, JMCB 2019).
2) 5 year, 5 year forward expected inflation. This is what markets expect inflation will be, in 5 years’ time, for the next 5 years. This measure tracks long run inflation expectations and removes the effects of short run shocks. In US data, big changes in inflation have been caused by unanchored long run inflation expectations, not by short run shocks to demand (see e.g. Hazell, Herreno, Nakamura & Steinsson 2021). So, if inflation is going to rise by a lot, long run inflation expectations are a good leading indicator.
For now, neither measure is high by historical standards but of course that could change. I hope some of this is interesting, anyway.