The 100 degrees outside tire temp means that the inside is even warmer.
I played with my pressure temp spreadsheet and made picture to make it clear, so first 2 pictures and then the explanation.
You filled 50 psi when outside temp was 70 degr F.
This becomes 61,6 psi when inside tire temp is 165 degr F, when air inside tire is absolutely dry, wich it never is, but will asume it. So when outside 65 degr , inside tire is as I once determined when cold pressure is right for the tire and loading, 112 degr F so 47 degr F higher then outside.
This means for 100 degr outside , inside tire temp normal and save would be 147 degrees.
filled in at the bottom 165 degrees inside tire temp to give the pressure you mention and so about 20 degr F to high.
This can also be courced by severe braking, or sunshine on tire, but then it would be only for the tires in the sun.
If it is courced only by the heat-energy produced by the tire driving at your speed so bending of rubber, it means that your cold pressure is to low and you need higher pressure to bring the warm inside tire temp down, wich is the main goal, and GmandJM already wrote that inside tire temp is more important to monitor then pressure.
If you can produce accurate determined loads on seperate tires, second bes axles, and give maxload and speedcode of tire ( AT pressure of 50 psi=C-load alreasy given) , I can calculate a save pressure for your situation.
Might give that 10 psi higher cold pressure , so 60 psi is needed and most tire-makers allow this.
If that 60 psi is the pressure that gives when filled cold at 70 degrees an inside tire temp of 70+47= 117 degr F only by the heat generated by bending of rubber, wich I state to be ok, the pressure then would rise to 66.6 psi , so about 11% higher
pressure.
The 212 degr F in the middle is boiling point of water , wich some tires inside air can incidentially get by the heat of brakes when severe braking when deschending from the mountains.
Water in tire, can rise the pressure from 70 to 165 degrees by maximum 5 psi, but filled at 70 degr 100% humidity outside air will give no more then 3 psi extra rising of pressure at this temp rising.
So inside tire temp can be lower then my given 165 degrees when expected unintended amount of water in tire.
