The Ramanujan–Kolberg identity conjecture for multipartitions

Let $p_t(n)$ denote the number of $t$-multipartitions of $n$. Let $a$ and $t$ be odd integers, with $3\mid t$ whenever $3\mid a$. Define $b$ by

$$b\equiv \frac{t}{3}\cdot 8^{-1}$$

when $3\mid t$, and by $b\equiv24^{-1}\pmod a$ otherwise, and set

$$k=\left\lceil\frac{t(a^2-1)}{24a}\right\rceil.$$

Here $f_d$ denotes the standard eta-product factor used in the source, and $\epsilon^t_{a,d,j}$ are coefficients in $\{0,1\}$. Ramanujan–Kolberg identity conjecture. There is a suitable choice of the $\epsilon^t_{a,d,j}$ such that $\epsilon^t_{a,1,0}=1$, $\epsilon^t_{a,d,j}=0$ whenever $at/d-24j<0$, and

$$q^k\sum_{n=0}^{\infty}p_t(an+b)q^n\equiv\sum_{d\mid a}\sum_{j=0}^{\lfloor k/d\rfloor}\frac{\epsilon^t_{a,d,j}q^{dj}}{f_d^{at/d-24j}}.$$

The conjecture predicts that these subprogression generating functions lie in a particularly structured, relatively low-dimensional space of modular forms. The supplied text says Chen proved it for, among other cases, all primes $a\geq3$, so the general conjecture remains open.