Atkin–O'Brien partition congruence conjecture for powers of 13

Let $P(N)$ be defined by $P(N)=p(n)$ when $N=24n-1$, and $P(N)=0$ if $N<-1$, if $N\not\equiv -1\pmod{24}$, or if $N$ is nonintegral. Let $\left(\frac{a}{b}\right)$ denote the Jacobi symbol. Atkin–O'Brien partition congruence conjecture. Let $\alpha\geq 1$ and let $p\neq 13$ be a prime with $p\geq 5$. Then there exists a constant $k=k(p,\alpha)$ such that, for all $N$,

$$P(p^2\cdot 13^\alpha N)-\left\{k-\left(\frac{-3\cdot 13^\alpha N}{p}\right)p^{-2}\right\}P(13^\alpha N)+p^{-3}P\left(\frac{13^\alpha N}{p^2}\right)\equiv 0\pmod{13^\alpha}.$$

The paper states that this conjecture was made by Atkin and O'Brien and proves it, so the congruence is no longer open.