Conjecture on bounds for normalized Bessel eigenfunction constants

Let $B$ be the unit disk, let $(\rho_p)_{p\in\mathbb{N}^*}$ be the positive roots of the Bessel function $J_0$, and define

$$e_p(\boldsymbol{x})=C_pJ_0(\rho_p\lvert\boldsymbol{x}\rvert),$$

where $C_p$ is chosen so that $\lVert e_p\rVert_{H^1_{\mathrm{rad}}(B)}=1$, equivalently

$$C_p=\frac{1}{\sqrt{\pi}\rho_p\lvert J_1(\rho_p)\rvert}.$$

Bessel normalization conjecture. For every $p\in\mathbb{N}^*$,

$$\frac{1}{\sqrt{2\pi p}}\leq C_p\leq\frac{1}{\sqrt{2\pi(p-1/4)}}.$$

The paper notes that the bounds are consistent with the known asymptotic expansion $C_p=1/\sqrt{2\pi p}+O(p^{-3/2})$ and reports strong numerical evidence, but states that the conjecture seems hard to establish and that no proof was found in the literature.