difficult flattening

[Old Murgh]

Since you found it hard to imagine my geometry, I thought I'd give it another try with more visual effort.
This is the simple model and its reference card. A geometrical impossibility? Possibly.


By the eye, I can get fairly close, while retaining nice and perfect vertex coordinates. Simplicity is the guiding star here.

But obviously one can only get so close and the pentagons will not be perfectly flat. Enter the need for the "anchor vertex" command when performing the flatten.

These are the desired goals. Seeing this does it change the way you understood my initial question?

I have tried my best to tug and pull as you described below, but being a three-pronged problem I get so far outside my goals that I keep starting over again.

Seeing this now, is there maybe an altogether different way you would have tackled arriving at this simple design? Since I started this shape with a slice of a perfect dodecahedron I know I had perfect pentagons initially, but somewhere despite my best efforts, I lost it.

Or am I wrong in thinking it's possible using Wings?