@MarcoGolla Maybe I missed that: where do we require $\partial u$ to be null-homologous? I was modeling after Figure 1, which also appears to traverse one of the generating curves. The winding around the cylinders was just fanciful, in any event! If I were to connect the points using straight lines (smoothed at corners), what is the obstruction to a curve connecting $z$ to $y$? I think that's my biggest concern here: I can't see when there are obstructions to existence of curves between generators.

@MarcoGolla To be clear: if they’re showing invariance and such, you’d assume they had already defined the objects well enough to draw pictures and see curves. That’s what I meant. Let me be a little more clear! Here is a stabilization of S^3 for simplicity. There are no disks starting at z in this picture. What is wrong with the “holomorphic curve” I drew in the corresponding cylindrical version? Picture

@MarcoGolla Sorry for the delay! I have looked through it. I haven't fully internalized all the details, but I don't think this quite does what I was hoping for: I was looking for clarification on how the holomorphic curves should look in the cylindrical version. I suppose I could pull back the disks in OS' version using this correspondence, but that seems a bit complicated, especially considering how late in the paper it comes?