(I think the problem is triangulation)
Triangulation sometimes has problems with concave shapes, and I believe this is caused by triangulation (without overlaps) being much more difficult in 3d than 2d. But the problem shouldn't really exist (at least in my case) because the triangulation would work fine using 2D triangulation by projecting the points to 2d along the average normal of the face (it bends less than 90 degrees, so 2D triangulation would be perfect).
Open the project, the shape is completely wrong on the flat faces when using the curve modifier. It's supposed to be a concave crescent shape, instead of the convex hull sorta shape it has now.