Conjectural description of maximizers for the maximum product of distances

Let $n\ge 3$. Let $P=\\{z_1,\dots,z_n\\}\subset\mathbb{C}$ be an $n$-gon with $\operatorname{diam}(P)\le 2$ that attains $\Delta(P)=\Delta_{\max}(n)$.

Maximizer structure conjecture. The following properties hold:

1. If $n$ is odd, $P$ is a regular $n$-gon.
2. If $n$ is even, $P$ has an axis of symmetry. Moreover, if $6\mid n$, then $P$ is invariant under rotation by $2\pi/3$, and hence has dihedral symmetry compatible with this $120^\circ$ rotational symmetry.
3. If $n$ is even, the diameter graph of $P$ is obtained from a cycle $C_{n-3}$ by attaching three pendant edges to vertices of the cycle.

This conjecture refines the structural restrictions and numerical evidence developed in the paper, including the observed behavior for small $n$ and explicit asymptotic constructions. The proposed description remains conjectural because the extremal configurations for finite $n$ are difficult to determine; proving these properties would give a substantially sharper characterization of the maximizers.