Generalized Hurwitz conjecture for zeros of Bessel function derivatives

Let $n\in\mathbb{N}_0$. For a real parameter $\nu$, write $J_{\nu}^{(n)}$ for the $n$th derivative of the Bessel function of the first kind with respect to its argument. Generalized Hurwitz conjecture. If $s$ is a nonnegative integer and

$$n-2s-2<\nu<n-2s-1,$$

then $J_{\nu}^{(n)}$ has $4s+2$ complex zeros, of which two are purely imaginary. If $s$ is a positive integer and

$$n-2s-1<\nu<n-2s,$$

then $J_{\nu}^{(n)}$ has $4s$ complex zeros, of which none are purely imaginary. This would provide a complete analogue of Hurwitz's theorem for derivatives of Bessel functions; the claim is posed as a question in the paper, and its resolution is not supplied.