Symmetry conjecture for optimal even small polygons

Let $n\geq 6$ be even, and let $A_n^*$ denote the maximal area among all small $n$-gons. An optimal $n$-gon is a small $n$-gon whose area is $A_n^*$. Its diameter graph is the graph whose vertices are the polygon's vertices, with edges joining pairs at distance equal to the diameter, and a pendant edge is an edge incident to a vertex of degree one. Symmetry conjecture. For even $n\geq 6$, an optimal $n$-gon has an axis of symmetry corresponding to the pendant edge in its diameter graph. The maximal-area problem for even small polygons is not fully resolved; the conjecture proposes additional symmetry for optimal polygons beyond the known structure of their diameter graphs.