CM placement conjecture for ramified quadratic extensions at 7

Let $E/\mathbb{C}_7$ be an elliptic curve with complex multiplication, and define

$$j_0(E)=j(E)-1728.$$

CM placement conjecture at 7. If

$$\operatorname{End}(E)\otimes \mathbb{Z}_7\cong \mathbb{Z}_7[\sqrt{-7}],$$

then $v_7(j_0(E)^4-7^4)>4$; if

$$\operatorname{End}(E)\otimes \mathbb{Z}_7\cong \mathbb{Z}_7[\sqrt{-21}],$$

then $v_7(j_0(E)^4+7^4)>4$.

The conjecture is proposed as part of a moduli-theoretic description of the components in the stable model of $X_0(125)$ and is supported by computations, but no proof is given.