Here's the thing. It's the inner tube that holds the air pressure - not the tire and not the rim. Something in the tire or the rim may be causing a puncture in the inner tube but its the tube alone that holds the air.
You haven't found where the air is leaking from yet. Finding that will greatly help your search for what's causing the loss of air pressure.
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My greatest fear is all of my kids standing around my coffin and talking about "how sensible" dad was.