RPB... he's always maintained that he does NOT advocate rolkur.
And no, LDR is NOT the same thing if ridden correctly - not just rolkur with a different name.
In a 'true' LDR, the horse is ridden with a deeper neck and softly asked to come a little behind the vertical, to assist in picking up the back and stretching the neck from the wither. There is no backwards force on the horse's jaw, and generally ridden with very light contact. This can be very useful for a horse that tends to brace through the neck, or one that is highly reactive. I will put my horse in LDR with no hesitation when I feel that he's going to blow up - he's extremely reactive and spooky, and to bring him a little deep is the 'safe' position for both of us until he settles.
If Carl's horses were rolkured, that would not be so beautifully up to the bridle. There is no pulling action on them at all. Compare how Carl/Charlotte's horses work compared to Adelinda's 'Parzival', or Partrik's 'Scandic', who are publically trained in rolkur.
I suspect Carl would be quite offended to think he was being accused of employing and advocating rolkur.