Fourier-critical-point conjecture for the auxiliary Bessel derivative function

Let $n\in\mathbb{N}_0$, let $\nu\in\mathbb{R}$, and define the real entire function

$$f_{\nu,n}(x)=\sum_{m\geq0}\frac{\Gamma(\nu+2m+1)}{\Gamma(\nu+2m-n+1)\Gamma(\nu+m+1)}\frac{x^m}{m!}.$$

A Fourier critical point of a real entire function is a point at which some derivative has a critical zero, as defined in the paper. Fourier-critical-point conjecture. If $s$ is a nonnegative integer and

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

then $f_{\nu,n}$ has exactly $s$ Fourier critical points and one positive real zero. If $s$ is a positive integer and

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

then $f_{\nu,n}$ has exactly $s$ Fourier critical points and no positive real zeros. The conjecture is motivated by the Fourier-critical-point proof of Hurwitz's theorem and would imply affirmative answers to the two clauses of the generalized Hurwitz question above; the paper records it as an open problem.