Max inflation stamped on the sidewall is for cold tire inflation. On the Yoko AVID H4S, should be 44PSI. Depending on conditions at hand - it is not unusual for the tire pressure to increase several PSI when up to operating temps. In some cases, overinflation is as bad as underinflation - the correct tire pressure is dependant on vehicle's suspension, driving conditions, and load capacity.
Since my driving is usually on decent highways (toll roads) over a relatively good distance on a slightly lowered suspension and slightly higher spring rates than OEM - I can run the higher pressures and be OK. Not to say that all tires can be inflated that high - just depends on the tire and experience you have at the time. There are a few tires that I run as high as 50PSI - 54PSI, because I know they can take it. In tire R&D - we sometimes run tire pressures as high as 200-300PSI with 2000-3000 pounds of force at 85-115MPH and run the tire to failure. I've seen tires blowup after a few minutes of this torture - others can run for days/weeks. Guess which one's I prefer to run on my personal vehicles .
You also brought up a very good point of tire pressure variance between axles. Bitter makes a good point about tire pressure and vehicle dynamics - depending on you driving conditions (ie. autocross or just like to drive "spiritedly" - running heavier spring rates or higher tire pressures in the rear axle of a FWD vehicle usually results in reduced control and tendancy for the back end to snap around in an emergency maneuver. Lowering the rear tire pressure or going with a softer spring rate will help keep the rear end following the fronts.
On long highway hauls where I might not touch the brakes or make big steering corrections - higher pressures will enhance tire life, keep tire temps and tire shape in check, and maximize fuel economy/minimize rolling resistance. Caveat is that I know that running higher pressures will sacrifice ultimate traction, potential for more damage from a very severe road hazard due to a blowout, and braking will suffer. But if I need that added traction and bigger contact patch - I know that I can drop the pressure, easiest way to get that back.
That's what makes it so hard - there is no "ideal" tire pressure for everyone. Good way to find out is to treat the tire pressure on the vehicle placard (~30-32PSI) as the absolute minimum tire pressure and treat the max cold inflation on the sidewall of the tire as the maximum. Fill the tire up to maximum tire pressure drive a little bit, then drop it in 2PSI steps until you hit a level where ride and performance are acceptable.