Kanade–Russell's recurrence-generation conjecture for cylindric Kanade–Russell identities

Let $m\geq5$ be a modulus of the form $m=3k+\{-1,0,1\}$ with $k\geq3$, and let $S_m(\rho\mid\sigma)$ be the multiple-sum functions indexed by integer vectors $\rho,\sigma\in\mathbb Z^{k-1}$. Let $R_1^{(i)}(\rho\mid\sigma)$ and $R_2^{(i)}(\rho\mid\sigma)$, for $1\leq i\leq k-2$, together with the applicable relations $R_3(\rho\mid\sigma)$ and $R_4(\rho\mid\sigma)$, denote the functional relations displayed in the source. Kanade–Russell's recurrence-generation conjecture. In each modulus $m\geq5$, the relations $R_1$–$R_4$ are enough to prove the recurrences needed for the proof of the cylindric Kanade–Russell conjecture. The paper reformulates this as an ideal-membership problem and studies it computationally; the general assertion remains open in the stated context.