The maximal edge-length sum conjecture for full grid lattice polygons

Let $a(n)$ be the maximum of the sum of the squares of the edge-lengths over full grid lattice polygons with $n$ vertices, and let $\alpha(n)$ be the explicit lower-bound function defined in the preceding result. Maximal edge-length sum conjecture. For $n\geq 9$,

$$a(n)=\alpha(n).$$

The conjecture asserts that the construction achieving $\alpha(n)$ is optimal for every $n\geq 9$; the supplied text gives no resolution, so the claim remains open.