Optimal convergence rate for the modular distribution of minimal residues

Let $F_N(r)$ denote the empirical distribution of $q_{\min}\bmod \ell$, let $\pi_{\mathrm{perm}}(r)$ be the permitted-residue distribution, and let $C_{\mathrm{opt}}(k,\ell)$ be a constant depending on $k$ and $\ell$. Optimal-rate conjecture.

$$\limsup_N \sqrt N\max_r\left|F_N(r)-\frac{\pi_{\mathrm{perm}}(r)}{\ell-1}\right|\le C_{\mathrm{opt}}(k,\ell)\approx0.5.$$

The proposed constant is independent of $k$ and is roughly five times smaller than the triangle-inequality bound; proving this would require a Cauchy--Schwarz refinement.