The full-rank conjecture for the σ(N,l)-matrix

For $N\geq 2$, let $t$ be the number of positive divisors of $N$, let $g_1,\ldots,g_r$ be the generators specified in the surrounding setup, and let $\sigma_{N,l}$ be the associated quantities for divisors $l$ of $N$ with $1<l\mid N$. Full-rank conjecture.

$$\operatorname{rank}\left(\sigma_{N,l}(g_j)\right)_{1\leq j\leq r,\\,1<l\mid N}=t-1.$$

This conjecture concerns the rank governing the image of the relevant character map and would provide information needed for larger $N$. The source gives no proof or disproof, so its status remains open.