The quadratic convergence conjecture for the decreasing threshold sequence

Fix $\kappa\geq4$ with $\kappa\in\mathbb{N}_e$, and let $\{\mathcal{E}_n\}_{n=0}^{\infty}$ be the strictly decreasing sequence from the stated theorem, with limit $\inf J_2=\cos^2(\pi/\kappa)$. Quadratic-convergence conjecture. There is a constant $c(\kappa)$ depending on $\kappa$ such that

$$\mathcal{E}_n-\inf J_2=\frac{c(\kappa)}{n^2}+o\left(\frac1{n^2}\right),\qquad \forall\,\kappa\geq4,\ \kappa\in\mathbb{N}_e.$$

The conjecture is based on numerical solutions; the supplied text gives no resolution.