Sufficiency of the R1--R4 relations for the recurrence proofs

Let $S_m(\rho\mid\sigma)$ denote the finite-level $(z,q)$ sums, and let $R_1$--$R_4$ be the relations listed for the relevant moduli. The R1--R4 sufficiency conjecture. In each modulus, the relations $R_1$--$R_4$ are enough to prove the required recurrences. This asserts that the displayed relations form a complete proof mechanism for the recurrence systems in every modulus treated. The passage explicitly says that a full proof is not currently available, so the conjecture is open.