Limit behavior of distinguished angles at the Sobolev critical exponent

Let $\overline{\Theta}$ be the angle from Corollary B(iv), and let $\Theta^*$ be the angle from Theorem C, for the Emden–Fowler problem on a spherical cap. Critical-exponent limit conjecture. If $N\ge 4$, then

$$\overline{\Theta}\to 0\quad\text{and}\quad\Theta^*\to 0\qquad (p\downarrow p_{\rm S}).$$

If $N=3$, then

$$\overline{\Theta}\to\frac{\pi}{2}\quad\text{and}\quad\Theta^*\to\frac{\pi}{2}\qquad (p\downarrow p_{\rm S}).$$

This concerns the change in solution structure as the exponent approaches the Sobolev critical exponent from above; the supplied text presents it as a conjectural statement and gives no resolution.