Bárány's conjecture on extrema and Turán bounds for Bessel functions

Let $u>0$, and let $J_\nu$ be the Bessel function of the first kind of order $u$. Write $j_{\nu,n}$ for its positive zeros, set $\Xi=\{j_{\nu,n}:n\geq 1\}$, and define

$$\Phi_\nu(x)=\frac{J_\nu^2(x)-J_{\nu-1}(x)J_{\nu+1}(x)}{J_\nu^2(x)}$$

for $x\in(0,\infty)\setminus\Xi$. The roots of $\Phi_\nu'$ are denoted by $\alpha_{\nu,n}$.

Bessel-function conjecture. The equation $\Phi_\nu'(x)=0$ has infinitely many roots, with

$$\alpha_{\nu,n}\in(j_{\nu,n},j_{\nu,n+1})$$

for every $n\geq1$. Moreover, if $\beta_{\nu,n}=\Phi_\nu(\alpha_{\nu,n})$, then

$$J_\nu^2(x)-J_{\nu-1}(x)J_{\nu+1}(x)>\beta_{\nu,n}J_\nu^2(x)$$

for $x\in(j_{\nu,n},j_{\nu,n+1})$, the sequence $\{\beta_{\nu,n}\}_{n\geq0}$ is strictly increasing, and $\alpha_{\nu,0}=j_{\nu,0}=0$ with $\beta_{\nu,0}=\Phi(\alpha_{\nu,0}^+)=1/(\nu+1)$.

This conjecture proposes a sharp interval-by-interval improvement of Szász's Turán inequality, with the constants determined by the interior extrema of the normalized Turánian. Its resolution is not established in the supplied text.