The construction lines will not be very meaningful unless you study the picture below.
The “horizontal” construction lines are cut off so that one can start with a smaller patch (pictured) using NetworkSrf (tangency at top & bottom). See which exactly which curves to select for this first surface in the picture.
(I had to use a “pipe-cut fillet” on the upper edge, hence the narrow surface on top, plus a cut & patch on each side due to a small irregularity).
After the first patch, the rest of the surface can be made in one shot. All the other construction lines to be selected. Use NetworkSrf for both surfaces because of the horizontal lines, which Sweep2 won’t accept.
On the last big surface I used the first surface’s edges instead of the two vertical construction curves shown in the picture.
As mentioned earlier, I rounded off by cutting out the upper edge with a pipe and did a BlendSrf. That resulted om some small pieces as well (watch out when exploding) which I didn’t bother fixing.
Almost there, but only almost. So I’m looking for a more systematic and quicker way.