Anecdotally, I've had tires that were 3-4 lbs lower than the others and was too busy to fix it for a couple of days. The lower pressure tires never caught up to the higher pressure tires no matter how long I drove (maybe 40 minutes on a highway commute).Also we can do a thought exercise: placard pressure 45, Set one tire to 41, one at 45, and another at 49. Day high is 40C (104F). After an hour or two of highway speed driving, All three tires will be about 53/54 psi as they reach thermal equilibrium**. However the one that started at 41 will be much hotter than the one at 49.
** Tires typically have an 8% pressure increase when driving at North American highway speeds, but might be as high as 15% if there is a big temperature difference between morning and afternoon temperatures. In other words, they're all going to reach the same pressure eventually, so it's better to start high and avoid the excessive heat buildup which is really bad for tires.
I also don't understand how they can reach a pressure equilibrium given one tire has less air. If the temperature comes from flexing and the flexing comes from pressure and the cooling is the same on the tires (because they are on the same vehicle), then less air will mean less pressure which will mean more flex leading to higher temp. That I can see.
But if it also causes the pressures to eventually equal out then the pressure is the same in both tires which means the flexing is the same, which means the induced temp should be the same, which means the tire with less air in it will have a lower pressure. The two fight against each other. In each tire the temp would reach an equilibrium with the pressure in that tire, but the different tires should not reach the same pressures as a result.
I can see the lower pressure tire running hotter, but I don't see the pressure catching up ever.