Bend surfaces at defined point (or rather segment)

Good morning!
I’m having trouble with creating a conic shape that should be connected in certain points. Basically, it follows the idea of Kirigami cuts. Now I want to create a 3D-Model instead of just 2D-Lines.
So I took my list of lines, offset each of them with the correct distance and connected the pairs by using the Weave-Component and after that, Boundary Surface. All of this works fine. But: How can I make them act like they’re connected on certain segments?
I attached a file where the generated surfaces and the list of points I used are visible. My goal would be to have every other surface connected to the next by a bridge (located around the points), like in this picture:

Does anyone here have an idea how I can accomplish this? Thanks in advance!

conic (7.6 KB)