The general-field moduli conjecture for intrinsic torsion subgroups

Let $k$ be any field and let $E$ be an elliptic curve over $k$ with

$$E(k)_\mathrm{tors}=\langle P\rangle\simeq\mathbb{Z}/N\mathbb{Z}.$$

Write $E(k)_\mathrm{tors}^\mathrm{is}$ for the intrinsic subgroup, and let $H$ be the subgroup associated with a twist $X_H$ of $X_{1,0}(N,MN)$ appearing in the stated family. General-field moduli conjecture. The point $(N/M)P$ lies in $E(k)_\mathrm{tors}^\mathrm{is}$ if and only if

$$\operatorname{im}\rho_E\leq H.$$

Equivalently, the moduli problem for elliptic curves with torsion $\mathbb{Z}/N\mathbb{Z}$ and intrinsic subgroup $\mathbb{Z}/M\mathbb{Z}$ is solved over a general field by the curves $X_1(N)_M^\zeta$. The authors establish the corresponding result over fields contained in $\mathbb{C}$, but their methods do not prove it over general fields; the claim is presented as likely, at least when $\operatorname{char}(k)\nmid N$.