The partition characterization of the Andrews–Gordon companion identities

Let $r\geq 2$ and $1\leq i\leq r$. Let $\mathcal{C}_{r,i}$ be the set of partitions $\lambda=(\lambda_1,\dots,\lambda_s)$ with at most $i-1$ parts equal to $1$ and satisfying either $N_{r,i}(\lambda)<r-1$, or $N_{r,i}(\lambda)=r-1$ and

$$s\leq \sum_{j=1}^{r-1}p_{i,j}(\lambda)-(r-i).$$

For a nonnegative integer $n$, let $C_{r,i}(n)$ be the number of partitions of $n$ in $\mathcal{C}_{r,i}$; let $T_{r,i}(n)$ and $E_{r,i}(n)$ denote the corresponding quantities defined in the paper. The partition characterization conjecture. For every nonnegative integer $n$,

$$C_{r,i}(n)=T_{r,i}(n)=E_{r,i}(n).$$

This conjecture gives a combinatorial interpretation of the Hilbert-series coefficients arising from the commutative-algebra approach to companions of the Andrews–Gordon identities. Its status is not resolved in the supplied source material.