The discrete Fourier-transform feasibility conjecture

Let $s\in\{+,-\}$, let $q\geqslant 1$, and let $\mathbb{A}_s^{\textup{disc}}(q)$ be the discrete feasibility constant defined for even functions on $\mathbb{Z}_{2q+1}$. A pair $(k,q)$ is $s$-feasible when there is an admissible function with $k_{sf}\leqslant k$.

Discrete Fourier-transform feasibility conjecture. If $(k,q)$ is $s$-feasible, then $(k+1,q)$ and $(k,q-1)$ are $s$-feasible. The function $q\mapsto\mathbb{A}_s^{\textup{disc}}(q)$ is non-decreasing, its range contains all integers $k\geqslant2$ for $s=+1$ and all integers $k\geqslant3$ for $s=-1$, and

$$\lim_{q\to\infty}\frac{\mathbb{A}_s^{\textup{disc}}(q)}{\sqrt{2q+1}}=\mathbb{A}_s(1),$$

where $\mathbb{A}_s(1)$ is the corresponding continuous one-dimensional constant.

This conjecture asserts that the discrete feasibility patterns converge to the continuous sign uncertainty principles. The source reports numerical evidence but no proof.