Non-negativity conjecture for the inverse matrix in the effective case

In the effective case $|G_0|=1$, let $g_1=i_0$ and consider the matrix

$$\begin{bmatrix} l_{g_1}^{(g_1)}&l_{g_2}^{(g_1)}&\cdots&l_{g_s}^{(g_1)}&l_{0}^{(g_1)} \\ \vdots&\vdots&&\vdots&\vdots\\ l_{g_1}^{(g_s)}&l_{g_2}^{(g_s)}&\cdots&l_{g_s}^{(g_s)}&l_{0}^{(g_s)} \\ 1&0&\cdots&0&0 \end{bmatrix}.$$

The non-negativity conjecture. The entries of the inverse of the above matrix are non-negative integers. This condition is presented as a sufficient condition ensuring that the corresponding $m_g$ are all non-negative integers in the effective case.