Monotonicity conjecture for positive zeros of Bessel function derivatives

For $n\in\mathbb{N}_0$ and $m\in\mathbb{N}$, let $j_{\nu,m}^{(n)}$ denote the $m$th positive zero of the $n$th derivative $J_{\nu}^{(n)}$ of the Bessel function of the first kind. Monotonicity conjecture. For fixed $n\in\mathbb{N}_0$ and $m\in\mathbb{N}$, the map

$$\nu\longmapsto j_{\nu,m}^{(n)}$$

is increasing on $(n-1,\infty)$. For $\nu>n-1$, the zeros of $J_{\nu}^{(n)}$ are real, and the conjecture would extend known monotonicity results for $n\in\{0,1,2,3\}$ to every nonnegative derivative order. The paper presents this as an unresolved question motivated by applications to starlikeness and close-to-convexity.