Redheffer-type bounds for normalized Bessel functions

Let $u>-1$, let $j_{ u,1}$ denote the first positive zero of the Bessel function $J_ u$, and let $\mathcal{J}_ u$ denote the normalized Bessel function used in the source. Define

$$\vartheta_ u=\frac{j_{ u,1}^2}{4(\nu+1)(\nu+2)},\qquad \eta_ u=1.$$

Redheffer-type bounds conjecture. For $|x|<j_{ u,1}$,

$$\left(\frac{j_{ u,1}^2}{j_{ u,1}^2-x^2}\right)^{\vartheta_ u}\leq\frac{\mathcal{J}_{\nu+1}(x)}{\mathcal{J}_\nu(x)}\leq\left(\frac{j_{ u,1}^2}{j_{ u,1}^2-x^2}\right)^{\eta_ u},$$

where $\mathcal{J}_\nu$ is the normalized Bessel function of the first kind. The constants $\vartheta_ u$ and $\eta_ u$ are best possible. The bounds would follow from the proposed monotonicity of the coefficient sequence discussed immediately before the conjecture; that monotonicity was established only for $\nu=-\frac12$ in the source, so the claim remains open for general $\nu>-1$.