If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$ What is known about $$ \sum_{n=-\infty}^{\infty} J_n^4(x) ? $$ It is smaller than $1$, but is there a nice lower bound?

The paper here mentions the integral $\displaystyle\sum_{n = -\infty}^\infty J_n^4(x) = \frac{2}{\pi} \int_0^\frac{\pi}{2} J_0^2(2x \sin \theta) d\theta$ and the asymptotic $\displaystyle \sum_{n = -\infty}^\infty J_n^4(x) = \frac{1}{x\pi^2}(\log x + 5 \log 2 + \gamma) + O(x^{-\frac{3}{2}})$ as $x \to \infty$, and gives references.