It's a real effect. A twin-axle trailer (caravan) is a statically indeterminate system.We have a 3380kg ATM van and generally within 100 kg so around 3200 to3300 with ball weight around 260-270kg on our 180kw 2015.
I use 34 front and 46 rear on the Touareg AND 45 on front axle and 50 on rear axle of van.
It was noticeable the difference with different axle pressure on the van. Theoretically the van pressures were calculated using the weight and pressure of actual versus max of tyre
This is a result of a guy called Collyn Rivers on another forum suggesting working out theoretical and dropping front axle by a few psi and increasing rear by same amount. This is for both van and towing vehicle. Theory correct ??? Placebo effect ??? Who knows but it does seem to work for me.
When a fixed structure is supported at just two points (front and back), it is statically determinate. This is the case for a single-axle trailer. You can calculate the loads on it with simple high school physics. The weight borne by each support point (tongue and axle) is proportional to their distance from the trailer's center of mass. So you can shift the loading towards the front or the back only by moving the trailer's center of mass forward or backward. (As an engineer, you can also modify weight distribution by moving the support points forward or back, but that's not an option in an already-manufactured trailer.) Raising or lowering the front/back doesn't matter, at least not until you get to extreme angles.
A statically indeterminate system doesn't work like this. Moving the center of mass forward or back (or moving the support points forward or back) isn't the sole determining factor in how much weight gets put on each support point. The height of each support point also makes a huge difference. Higher supports end up bearing more weight. You can think of this in the extreme case where the tires on the front axle are so deflated that they no longer touch the ground. That axle is now supporting zero weight, while the rear axle supports all the weight (minus the tongue weight). Now reverse the pressures so the rear axle is underinflated, and the front axle supports all the weight while the rear axle supports none.
So yes, varying the front/rear axle tire pressure has a real effect on the trailer's weight distribution. So does the hitch height, which is why they say to always run twin-axle trailers with the proper hitch height so that it's level. That's vital for assuring load is distributed between the hitch and front/rear axles as intended. There was a post a few months back by a guy whose trailer dealer rigged it up wrong. The hitch height was way too low, causing the twin-axle trailer's rear axle to lift and support less weight. The trailer's front axle was bearing nearly all of the load, meaning the trailer's center of mass was further back relative to the wheels on the ground, causing it to sway.
This is less true for the tow vehicle (which only has two support axles), assuming you're using a simple tow ball (pivot joint) with downward tongue weight. In the U.S., it's popular to use a weight distribution hitch, which does not let the connection between trailer and tow vehicle pivot freely. That couples the trailer with the tow vehicle in pitch, creating a more complex statically indeterminate system (effectively a fixed structure with 4 support axles), allowing you to transfer the trailer's tongue weight to the front or back of the tow vehicle as desired (basically adjusting the pitch angle at which the tow vehicle and trailer meet).
The wiki article on this is a bit too technical. But you can read through it if you want to learn more and can follow the math.