Endpoint stability and instability for high-energy mode ratios

Let $m,n$ be the mode indices, let $P$ be the parameter, and let $\Theta_m$ and $\Theta_n$ denote the corresponding modes with energy $E_{\Theta_m}$. For $k\in\mathbb N$, assume

$$0\leqslant P<m^2<n^2.$$

Endpoint behavior conjecture. There exists $E_k>0$ such that if

$$\frac{n^2}{m^2}=(k+1)(2k+1)$$

and $E_{\Theta_m}>E_k$, then $\Theta_m$ is linearly unstable with respect to $\Theta_n$, whereas if

$$\frac{n^2}{m^2}=(k+1)(2k+3)$$

and $E_{\Theta_m}>E_k$, then $\Theta_m$ is linearly stable with respect to $\Theta_n$.

The surrounding results leave these endpoint ratios outside the open intervals where the high-energy behavior is determined. The conjecture predicts opposite stability behavior at the two endpoints.