Congruence conjecture for coefficients of classical modular polynomials

Let $ell$ be a prime, let $a_{m,n}$ be the coefficient of the classical modular polynomial $\Phi_ell(X,Y)$, and suppose $ell+1>m+n$. Define

$$c:=ell+1-m-n.$$

Congruence conjecture. The following divisibilities hold: if $ell\ne 2$, then $a_{m,n}\equiv 0\pmod{2^{15c}}$; if $ell\ne 3$, then $a_{m,n}\equiv 0\pmod{3^{3c}}$, and if additionally $ell\equiv 1\pmod 3$, then $a_{m,n}\equiv 0\pmod{3^{\lceil 9c/2\rceil}}$; if $ell\ne 5$, then $a_{m,n}\equiv 0\pmod{5^{3c}}$. These predicted congruences motivate the study of divisibility properties of modular-polynomial coefficients; the source presents them as a conjecture and gives supporting computations, while related special cases are proved later.