CM residue-disk placement conjecture for supersingular reductions

Let $A$ be a supersingular elliptic curve defined over $\mathbb{F}_p$, and let

$$i=i(A)=|\operatorname{Aut}(A)|/2.$$

For a CM elliptic curve $E/\mathbb{C}_p$, write $\bar E=A$ when its reduction is $A$, and suppose

$$p\mathrel{||}\operatorname{Disc}(\operatorname{End}(E)).$$

A uniformizer $u\in\mathbb{C}_p$ of the common image and maps $\lambda_i:\bar E_i\to A$ are as in the claim.

CM residue-disk placement conjecture. The $j$-invariants of all such curves $E$ lie in $2(p+1)/i$ residue disks inside the corresponding Atkin–Lehner circle, with $(p+1)/i$ disks for each ramified quadratic extension of $\mathbb{Q}_p$. Two such curves $E_1$ and $E_2$ lie in the same residue disk if and only if

$$\operatorname{End}(E_1)\otimes\mathbb{Z}_p=\operatorname{End}(E_2)\otimes\mathbb{Z}_p$$

and

$$\sigma_{\lambda_1}(u)\equiv\sigma_{\lambda_2}(u)\in\bar A_p.$$

The conjecture is intended to explain the geometric placement of CM $j$-invariants and to provide a general theory behind the prime-specific CM placement conjectures at $5$, $7$, and $13$. The paper states that it has evidence but no proof.