Monotonicity of zeros of derivatives of Bessel functions

Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the $n$th derivative of the Bessel function of the first kind $J_{\nu }\left(x\right)$ are all real when $\nu >n-1$. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every $n\in N$, the positive zeros of $J_{\nu }^{\left(n\right)}\left(x\right)$ are increasing functions of the parameter $\nu$, for $\nu \in (n-1,\infty )$. We provide two apparently distinct proofs of the conjecture.