cannot manually place some cones

[AnnieLocke]

Heya - I'm still new to cone placement and conformal mapping in general but figured I'd try asking here. I'd like to map this column mesh and have straight edge boundaries on the left and right on the flat mesh. I tried manually placing the cones but can only select some vertices and not others. This is the closest I can get to straight lines on the L/R. Any tips/thoughts?

file:
https://drive.google.com/file/d/1_jqVtFZKLREAlXKJD11Vp8hp-m-jbxFg/view?usp=sharing

[rohan-sawhney]

Here are two layouts I created. The first one has a single cone with angle pi while the second one has two cones with angles pi & pi respectively. Note that one of the holes appears in the center of the flattening, this happens because internally BFF triangulates the shorter hole(s) to get a mesh with disk topology.