Asymptotic conjecture for the first positive zero

Assume the extremal problem has a unique solution $\varphi_p$, and let $t_1=t_1(p)$ be its first positive zero. First-zero asymptotic conjecture.

$$\lim_{p\to\infty}t_1=\frac12$$

and

$$\lim_{p\to0^+}pt_1>0.$$

The source presents these as suggested asymptotics from its analysis.