The original chair has top, middle, and lower rails that connect the two backrest stiles.   In addition to providing additional stability, I think that a big reason for the existence of the middle rail is to provide a landing point for the vertical caning of the backrest.   If there were only a top and bottom rail, the natural path of the caning between the two rails would have gone way outside the curve of the stiles (green line below).   The middle rail would have provided an intermediate anchor point for the vertical caning to bring it closer to the desired curve of the backrest.

My middle rail needs to serve a similar purpose for the vertical warp cords of the Danish weave.  I used a scrap of white oak for a proof of concept for the idea I have for the rail.   Basically, the rail will have a series of holes along the front edge, each large enough to pass a pair of warp cords.  I spaced the holes so that the gap between adjacent pairs of warp cords would accommodate 5 wraps of cord.  I also make one hole large enough to accommodate 4 warp cords.  On the prototype, that hole is centered, but on the actual rail there would be one on each end of the rail (if I decide I want to have a border of 4 cords).

I put a 1/4" radius round over on the edges and a 1/8" radius round over inside the holes.  Here, the prototype rail is set inside the remnant of the scrap it came from just to provide more support for the router base.

This is roughly where the middle rail would be located on the stile.

Verifying the fit for two warp cords and 5 wraps.

Fully wrapped.

I used 1/4" crown staples the tack down the cord.  None of the fasteners would be visible on the finished chair.  Only the curved edge sides would be visible.

The spacing seems good on the back side of the rail.

And this is how it would look on the front side of the rail.  The vertical wraps will be a noticeable transition between the horizontal weft weaving above and below the rail.   Hopefully, it won't be too visually jarring.

After I got to this stage, I realized that I forgot to account for the additional interior slot that will be needed on each rail.  The slot provides an alternative path from top to bottom, which allows the warp pattern to be different on the back than on the front.  Below are pics from an earlier project where I used this technique.  The wider band of warp cords on the back provides a hiding place for knots when joining lengths of the weft cords while weaving the backrest.  I need to go back and look at the prototype rail and see how best to accommodate the slot and still be able to wrap the rail as shown above.