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The dealer (any tire change place) is not lying to you - the pressure will be high when checked at higher indoor temperature and will go down if the ambient temperature is much lower (in your case). Best time to check air pressure is in the morning before moving your car. When I had the snow tires mounted, I asked for 40 psi since it was much warmer in the garage. When I checked in the morning, the pressure was 36 psi which is what I wanted. Just remember PV=nRT. If the temperature goes down, your pressure will go down as well as explained by this ideal gas law.
That is correct. Also, after driving tire pressures will also be higher because of all the air moving around in the tire heats up. Thats more of a cause of inaccurate tire temps down here in South Florida.
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