This is my speculation and perhaps Michael Alden, who saw (and photographed) some of the Duke's wardrobe, might shed some light on this.
The Duke appears to have had peculiar ideas on underwear.

Note how a pair of specially designed cotton boxer shorts is buttoned to the trousers' waistband, like a full and detachable lining. I don't know whether these trousers are singular in displaying this feature. At any rate, it makes away with the need for underwear as a DISTINCT item, incorporating it into the trousers.
However, unlike in a pair of "normal" boxers, the sides cannot overlap to form a fly. To avoid direct contact between skin and trousers, a shirt with long tails is needed:
Even if the shirt has side vents (slits), the front halves are pulled back and tend to gap when walking. This is not a problem when you wear "full" underwear, but not desirable when there is nothing else but the shirt tails between you and the trousers' fly. Therefore, moving the front opening to a side solves the problem. Practically, the boxers' fly is moved to the shirt tails! If the tails are long enough, you can tuck them under the crotch in front and they will stay in place throughout the day. I hope my detective work makes sense...
In fact, when you wear a shirt with long tails (i.e. normal dress shirt length, not like many RTW shirts which have barely 10-15 cms below your trousers' waistband) and if the buttons come almost all the way down (as most shirtmakers make them), when you go to the bathroom you need to unbutton (or unzip) the trousers' fly, UNBUTTON AT LEAST ONE SHIRT BUTTON if it is in your way and only THEN you have access to your boxers' fly (worse if you wear tangas). If the tails are long enough that you can tuck them under so they won't gap, you can do away with these lower shirt buttons (as in the Duke's shirts pictured), which is a practical idea. If I am not wrong, the Duke's design is a good piece of clothing engineering.
The strip of cloth with buttonholes is rather high, but it can serve no other purpose than the one described by Marcelo (and this is not speculation, because contemporary dress shirts have it, too). It secures the position of the shirt, making sure there is no unsightly bulging in front and that the tails stay in place inside your trousers.