Judging by your captures and my experience I would suggest (as already stated by Tony and others) that there is nothing wrong with the lambda control system on this car and everything wrong with the cat that has been fitted each year.
I came across these denso Lambda sensors on toyotas years ago and was baffled then but came to realise (as aiden says) they work different to normal zirconia sensors. During warm up and cruise, periods of rapid changing conditions the sensor trace looked very much like a normal sensor, but during idle when warm, periods of slow less active conditions the lambda trace amplitude was much less, to the point where it could flatline. Most scantools will also show current readings in live data for lambda sensors.
So if he doesn't want to spend much let him fit another aftermaket cat and get another 12 months. Otherwise a gen cat will be required.
Mike
M.O.M. Miserable Old Manc
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