CM placement conjecture for ramified quadratic extensions at 13

Let $E/\mathbb{C}_{13}$ be an elliptic curve with complex multiplication, and define

$$j_0(E)=j(E)-5.$$

CM placement conjecture at 13. If

$$\operatorname{End}(E)\otimes \mathbb{Z}_{13}\cong \mathbb{Z}_{13}[\sqrt{-13}],$$

then $v_{13}(j_0(E)^{14}-13^7)>7$; if

$$\operatorname{End}(E)\otimes \mathbb{Z}_{13}\cong \mathbb{Z}_{13}[\sqrt{-26}],$$

then $v_{13}(j_0(E)^{14}+13^7)>7$.

The conjecture extends the proposed CM-point placement pattern to $p=13$ and is supported by examples, but the paper does not prove it.