Finite truncation conjecture for the Kanade–Russell recurrence ideal

Fix $m=3k+\{-1,0,1\}$ with $k\geq3$. Let $\mathcal S_m$ be the set of relations obtained from the coupled $q$-difference equations in terms of the functions $S_m(\rho\mid\sigma)$. For $N\in\mathbb N$, let $I_{m,N}$ be the ideal generated by 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 $\rho,\sigma\in[N]^{k-1}$. Finite truncation conjecture. There exists some $N\in\mathbb N$ such that

$$\forall h\in\mathcal S_m,\qquad h\in I_{m,N}.$$

This is a stronger, computation-oriented form of the ideal-membership conjecture: for each fixed modulus, all required relations should follow from a finite bounded collection of recurrence relations. The source explicitly treats this as an open problem and proposes linear-algebraic approaches.