Infinitely many Ramanujan-type congruences for 3- and 5-regular partitions

Let $b_k(n)$ denote the number of $k$-regular partitions of $n$, where $k\in\{3,5\}$, and let $m$ be a positive integer. A Ramanujan-type congruence is a congruence of the form

$$b_k(An+B)\equiv 0\pmod m$$

for suitable integers $A$ and $B$. The 3- and 5-regular partition congruence conjecture. For $k\in\{3,5\}$ and every positive integer $m$, there are infinitely many Ramanujan-type congruences modulo $m$.

The paper proves this assertion when $m$ is prime and $m\geq 5$; the conjecture extends the existence of infinitely many such congruences to every positive integer modulus.