Mossinghoff's symmetry conjecture for maximal-perimeter convex small polygons

Let $n=2^s$ with integer $s\ge 2$, and let $P_n^*$ be a convex small $n$-gon with the longest perimeter. Its diameter graph records the pairs of vertices at Euclidean distance one.

Mossinghoff's symmetry conjecture. The polygon $P_n^*$ has an axis of symmetry corresponding to one particular pendant edge in its diameter graph.

This assertion supplements the conjectured diameter-graph structure of optimal convex small polygons by specifying a geometric symmetry associated with one of its pendant edges. The source gives no resolution, so the conjecture remains open.