The infinite cylindric-partition generating-function hierarchy

For $k\geq 4$ and $k\geq i\geq 1$, let $CP_{(a,b)}(n)$ denote the generating function for cylindric partitions with profile $(a,b)$, as defined earlier in the paper.

Cylindric-partition hierarchy conjecture.

$$CP_{(2k-i,i-1)}(n)=\frac{1}{(q;q)_{2n}}\sum_{r=-\infty}^{\infty}(-1)^r q^{\frac{r((2k+1)r+2k-2i+1)}{2}}{2n\brack n-\frac{(2k+1)r}{2}+(2k-2i+1)\frac{(-1)^r-1}{4}}_q,$$ $$CP_{(2k-i-1,i-1)}(n)=\frac{1}{(q;q)_{2n}}\sum_{r=-\infty}^{\infty}(-1)^r q^{r(kr+k-i)}{2n\brack n-kr+(k-i)\frac{(-1)^r-1}{2}}_q.$$

These formulas extend the generating-function identities established for smaller values of $k$, including the cases treated by direct computer algebra. General proofs are not known.