A theta-block crank conjecture for $(k+j)$-colored partitions

Let $k,j$ be positive odd integers. Let $q$ and $\tau$ be the variables in the generating function, let $\textbf{z}$ denote the crank variables, and let $\zeta$ denote the corresponding root-of-unity parameter. Choose $a_i$ to be consecutive even integers and $b_i$ to be consecutive odd integers. Define

$$C_{k,j}(\textbf{z},\tau):= \prod_{n=1}^\infty\frac{(1-q^n)(1+q^n)(1+\zeta^{\pm a_1}q^n)(1+\zeta^{\pm a_2}q^n)\cdot\ldots\cdot(1+\zeta^{\pm a_{\frac{j-1}{2}}}q^n)}{(1-\zeta^{\pm b_1}q^n)(1-\zeta^{\pm b_2})\cdot\ldots\cdot(1-\zeta^{\pm b_{\frac{k+1}{2}}}q^n)}.$$

The theta-block crank conjecture. The function $C_{k,j}(\textbf{z},\tau)$ defines a crank for the desired $(k+j)$-colored partitions and explains most of the Ramanujan-like congruences satisfied by their partition numbers. The author suggests that theta-block theory may establish this general crank construction, analogously to prior work on crank generating functions. The statement is not accompanied by a resolution, and the paper does not specify precisely which congruences are covered by “most.”