  In this paper I will establish a special case of a conjecture that
intertwines the deep diagonal pentagram maps and Poncelet polygons. The special
case is that of the 3-diagonal map acting on affine equivalence classes of
centrally symmetric octagons. This is the simplest case that goes beyond an
analysis of elliptic curves. The proof involves establishing that the map is
Arnold-Liouville integrable in this case, and then exploring the Lagrangian
surface foliation in detail.
