Isogeny invariance conjecture for valuations of elliptic-curve j-invariants

Let $\mathcal{O}_K$ be a discrete valuation domain with field of fractions $K$ and residue field $k$, and let $v$ be the normalized valuation on $K$. Let $E_1/K$ and $E_2/K$ be elliptic curves admitting an isogeny of degree $d$ from $E_1$ to $E_2$, where $d$ is coprime to $\operatorname{char}(k)$. Isogeny valuation conjecture. If $\operatorname{char}(k)=2$ and $0<v(j(E_1))<15v(2)$, then $v(j(E_1))=v(j(E_2))$; if $\operatorname{char}(k)=3$ and $0<v(j(E_1))<3v(3)$, then the same equality holds, and moreover, if $ell\equiv1\pmod3$ and $0<v(j(E_1))<\frac92v(3)$, it still holds; and if $\operatorname{char}(k)=5$ and $0<v(j(E_1))<3v(5)$, then $v(j(E_1))=v(j(E_2))$. This conjecture is stated as implied by the main congruence conjecture and connects modular-polynomial divisibility with the behavior of isogenous elliptic curves over discretely valued fields.