Kanade–Russell's ideal-membership conjecture for cylindric recurrences

Fix $m=3k+\{-1,0,1\}$ with $k\geq3$. Let $\mathcal S_m$ be the set of relations obtained by writing the coupled $q$-difference equations for the profiles $c$ with $|c|+3=m$ in terms of the functions $S_m(\rho\mid\sigma)$. Let $I_m$ be the ideal generated by the left-hand sides of the recurrence relations $R_1^{(i)}(\rho\mid\sigma)$, $R_2^{(i)}(\rho\mid\sigma)$, $R_3(\rho\mid\sigma)$, and $R_4(\rho\mid\sigma)$, with the applicable $R_3,R_4$ determined by $m$ modulo $3$. Kanade–Russell's ideal-membership conjecture. For every $h\in\mathcal S_m$, one has

$$h\in I_m.$$

This is presented as an equivalent reformulation of the recurrence-generation conjecture. The paper notes that the ideal has infinitely many nominal generators but that only finitely many $S_m$ functions occur for each fixed $m$; the resulting problem remains open in general.