Atkin's 13-adic congruences for normalized coefficients of the j-function

Let $j(z)=q^{-1}+744+196884q+\cdots$, write $j=\sum_n c_n(j)q^n$, and for a positive integer $m$ with $c_{p^m}(j)\neq0$ define

$$t_m(j,n)=\frac{c_{np^m}(j)}{c_{p^m}(j)}.$$

Atkin's conjecture. For any prime $p\neq13$ such that $c_{p^m}(j)\neq0$ for all $m\geq1$,

$$t_m(j,np)-t_m(j,n)t_m(j,p)+p^{-1}t_m(j,n/p)\equiv0\pmod{13^m},\qquad \forall m,n\geq1,$$

and

$$t_m(j,13n)-t_m(j,n)t_m(j,13)\equiv0\pmod{13^m},\qquad \forall m,n\geq1.$$

The source states that this conjecture was proved by Koike and Katz using the convergence of repeated applications of $U_{13}$ to a single $13$-adic Hecke eigenform. Related conjectures for primes $p\leq23$ were also established, so this claim is solved.