Permutation conjecture for modular-polynomial evaluations outside the invariant image

Let $N$ be the level parameter, let $\alpha$ be the invariant, let $\operatorname{image}\alpha$ denote its image, let $\Phi_m^{\alpha}$ be the modular polynomial with level structure, and let $\psi(m)$ be the degree parameter appearing in the construction. Permutation conjecture. For any positive integer $m$ prime to $N$, there exists a permutation $\sigma$ on $\mathbb{C}\setminus\operatorname{image}\alpha$ such that

$$\Phi_m^{\alpha}(z,Y)=(Y-\sigma(z))^{\psi(m)},$$

for every $z\in\mathbb{C}\setminus\operatorname{image}\alpha$. This conjecture describes the evaluation of the modular polynomials at points outside the image of the invariant, where all roots coincide according to a permutation of the complement.