The discrete negative sign uncertainty square-size conjecture

Let $\ell\geqslant3$ be an integer and let $\mathbb{A}_-^{\textup{disc}}(q)$ denote the discrete negative sign uncertainty constant.

Discrete square-size conjecture.

$$\mathbb{A}_-^{\textup{disc}}\left(\frac{\ell^2-1}{2}\right)=1.$$

The source reports numerical confirmation for every $\ell\leqslant100$ and suggests proving the matching lower bound. No proof for all $\ell\geqslant3$ is given.