The decreasing-limit conjecture for even-indexed upside-down wells

Let $\kappa\geq4$ be even and let $2\leq j\leq\lfloor\kappa/2\rfloor$ be even. Let $\{\mathcal{E}_n\}$ be the sequence constructed in the two stated theorems. Upside-down-well limit conjecture.

$$\mathcal{E}_n\searrow\cos^2\left(\frac{j\pi}{\kappa}\right).$$

The sequence is known in the cited theorems to be decreasing and to lie in the indicated interval; the asserted limiting value is not proved in the supplied text.