Unique maximization of the invariant for logarithmic pentagram spirals

Let ${\cal C}(n,k)$ be the space of normalized convex $n$-gons with the additional points defining a pentagram spiral, and let $Z$ be the invariant on this space. The logarithmic pentagram spiral is the distinguished point of ${\cal C}(n,k)$ represented by the corresponding spiral configuration.

Uniqueness conjecture. For any $(n,k)$, the invariant $Z$ is uniquely maximized, and has a unique critical point, at the point representing the logarithmic pentagram spiral.

This conjecture asserts that the logarithmic pentagram spiral is the unique variationally distinguished configuration in ${\cal C}(n,k)$, consistent with the observed contraction behavior of averaging iterates of the pentagram map.