The bottom-dissection reformulation of the Andrews–Gordon companion conjecture

Let $r\geq 2$ and $1\leq i\leq r$. Let $\mathcal{B}_{r,i}$ be the set of partitions whose $(i-1)$-bottom dissection consists of $i-1$ successive bottom squares followed by bottom rectangles, with all bottom rectangles above $B'_{r-1}$ empty. Let $B_{r,i}(n)$ be the number of partitions of $n$ in $\mathcal{B}_{r,i}$, and let $A_{r,i}(n)$ and $E_{r,i}(n)$ denote the associated quantities from the paper. Bottom-dissection reformulation. For every nonnegative integer $n$,

$$B_{r,i}(n)=A_{r,i}(n)=E_{r,i}(n).$$

The reformulation uses the equality $\mathcal{B}_{r,i}=\mathcal{C}_{r,i}$ to express the same companion identities through successive bottom squares and rectangles. The supplied source does not establish whether the conjecture has been resolved.