It seems the gap occurs only at the first vertex.
This could be addressed #IToo as a possible bug.
So as a workaround just add an extra vertex (it will be good to place it as the point where you expect the start of some default segment), set it to the smooth, and make it first. For the Rule>>Corners Type>>set to use just Corner.
Adaptive mode is only working when the segments are the same X size.
So, again as a workaround, you can nest one L1S Generator to act as a single segment which will be used as Default.
I`m not sure about the rest of the Style and how you want to align everything (bottom/default/top) in line but those are just quick tips. Sorry, I don`t have time for more at the moment.