Dear Ruth,
I am thinking of turning the drawdown - and the weaving structure - i.e. warp gets weft and weft gets warp. And the tieup must be reversed (rising gets sinking and sinking gets rising)
Maybe the weft threads that float in the original structure have to be supplementary warp ends.
I' d try this out theoratically in a weaving program by selecting and cutting the threading and pasting it in the weft, and selecting and cutting the treadling and pasting it in the warp area.
I hope this makes sense what I am trying to explain.
Erica