Stable supersingular-component conjecture for X_0(p^3)

Let $p$ be a prime, let $R_p$ be the base ring used for the semistable covering, and let $A$ be a supersingular elliptic curve defined over $\mathbb{F}_p$. Consider the Atkin–Lehner circle and too-supersingular disk associated with $A$, the maps $\pi_p$ and $\pi_{p^2}$, and the CM disks specified by the CM residue-disk placement conjecture.

Stable supersingular-component conjecture. There is a semistable covering of $X_0(p^3)$ defined over $R_p$ such that, for each such $A$, one connected component of the supersingular locus contains only the following: one component lying via $\pi_p$ over the Atkin–Lehner circle of $A$; two components $E_1$ and $E_2$, which are respectively $\pi_p^{-1}$ and $\pi_{p^2}^{-1}$ of the too-supersingular disk of $A$; and $2(p+1)/i$ components lying via $\pi_p$ over the CM disks. The intersections among these components and with the six ordinary components of $X_0(p^3)$ are as depicted in the source's figure.

This conjecture proposes the general stable-model configuration suggested by the computation for $X_0(125)$, including the number and incidence pattern of supersingular components. It is not proved in the paper.