Two encounters with the Tire Pressure Monitoring System on my 2018 A3 e-tron, neither a cause for concern:
1. See
https://static.nhtsa.gov/odi/tsbs/2020/ ... 2-0001.pdf which describes a Tire Pressure Monitoring System error warning on startup. I had been encountering this error message occasionally since purchase (new) in 2019. Took to the dealer in 2022 since the warranty is ending soon, and tire pressures tested normal. The mechanic found this Audi Technical Service Bulletin 2059234/1 dated April 10, 2020 that applies to the vehicle, and responds to the issues raised in this thread. There is no fix/repair required. The TPMS warning on startup does go away after a moment. All good.
2. I also encountered a continuing error message - !!! TPMS warning showing in the bottom row of the instrument display. This started while out driving, and remained until attended to. I tested the tire pressure, all normal so no urgency and waited until taking to dealer for 10,000km service (cumulative after 3 years, probably don't really need a vehicle). Again, the dealer tested air pressure and all normal. Accordingly, I manually reset the TPMS using the MMI. Since then, no further !!!TPMS warning. The owners manual does mention that the system can malfunction for a variety of reasons, eg. sudden change in external air temperature.
Tentative conclusion: the TPMS is poor technology that provides a false sense of security, it's better to check tire pressure regularly, and it's probably safe to ignore the TPMS error warnings.
Of course, if you do get a flat tire warning, pull to a stop when safe and check it out!