Regularity and minimal-growth conjecture for Fuchsian singularities

Let $X$) be the domain in Eq. (1), let $\hat X$ be its compactification, and let $\zeta\in\partial\hat X$. Assume that Eq. (1) has a Fuchsian type singularity at $\zeta$ and admits a global positive solution. A point $\zeta$ is regular for the equation if any two positive solutions in $\mathcal{G}_\zeta$ are comparable with respect to the ordering $\precsim$; a positive solution has minimal growth in a neighborhood of a set when it has the minimal-growth property used in the paper. Main conjecture. Then: (i) $\zeta$ is a regular point of Eq. (1); and (ii) Eq. (1) admits a unique global positive solution of minimal growth in a neighborhood of $\partial\hat X\setminus\{\zeta\}$. This conjecture is the paper’s main conjecture and is only partially answered there; the supplied text gives no complete resolution.