Kanade–Russell's cylindric partition profile conjecture

Let $k\geq 3$, let $c=(c_1,c_2,c_3)$ be a cylindric partition profile with $|c|+3=m=3k+\{-1,0,1\}$, and, using cyclic symmetries, assume $c_1\geq c_2,c_3$. Let $H_{(c_1,c_2,c_3)}(z,q)$ denote the associated generating function, and let $e_i$ and $\delta_i$ be the vectors defined by

$$e_i=(\underbrace{0,0,\dots,0}_{i},1,1,\dots,1),\qquad \delta_i=(\delta_{ij})_{1\leq j\leq k-1}.$$

For $m\in\{3k-1,3k,3k+1\}$, let $S_m(z;\rho\mid\sigma)$ be the corresponding multiple-sum functions defined for integer vectors $\rho,\sigma\in\mathbb Z^{k-1}$. Kanade–Russell's conjecture. If $c_2,c_3\leq k-1$, then

$$H_{(c_1,c_2,c_3)}(z,q)=\begin{cases}S_m(z;e_{c_2}\mid e_{c_3})-qS_m(z;e_{c_2-1}\mid e_{c_3-1}),&c_2,c_3>0,\\S_m(z;e_{c_2}\mid e_0),&c_3=0,\\S_m(z;e_0\mid e_{c_3})-q(1-z)S_m(z;e_0+\delta_0\mid e_{c_3-1}),&c_2=0,\ c_3\ne0. \end{cases}$$

This is the principal cylindric Kanade–Russell conjecture for the three-part profiles; the paper develops proofs for the modulo $11$ and $13$ cases, while the general fixed-$k$ assertion is the conjectural framework motivating those results.