The discrete Hankel-transform feasibility conjecture

Let $s\in\{+,-\}$ and $\nu\geqslant-\tfrac12$. For functions on $[q]=\{1,2,\ldots,q\}$, let $\mathbb{A}_s^{\textup{disc}}(q,\nu)$ be the discrete Hankel-transform feasibility constant, and call $(k,q)$ $(s,\nu)$-feasible when an admissible function has $k_{sf}\leqslant k$. Let $\{j_n\}_{n\geqslant1}$ be the positive zeros of $J_\nu$.

Discrete Hankel-transform feasibility conjecture. If $(k,q)$ is $(s,\nu)$-feasible, then $(k+1,q)$ and $(k,q-1)$ are $(s,\nu)$-feasible. The function $q\mapsto\mathbb{A}_s^{\textup{disc}}(q,\nu)$ is non-decreasing, and its range contains $\mathbb{N}\setminus[k_0]$ for some $k_0\geqslant1$. Moreover, if $\nu=\tfrac d2-1$ and $n_q=\mathbb{A}_s^{\textup{disc}}(q,\nu)$, then

$$\lim_{q\to\infty}\frac{j_{n_q}}{\sqrt{2\pi j_{q+1}}}=\mathbb{A}_s(d),$$

where $\mathbb{A}_s(d)$ is the corresponding continuous sign uncertainty constant.

This is the Hankel-transform analogue of the discrete Fourier-transform conjecture. The source gives numerical support, particularly in dimensions $2$, $8$, $12$, and $24$, but no general proof.