Strengthen the first q-supercongruence modulo [m]Φ_m(q)^2

Prove that the q-supercongruence in Theorem 1, namely the congruence for the parameterized sum involving [6n−1], q-shifted factorials with base q^3, and parameters m, t∈{1,2}, and s≤(tm+1)/3 satisfying m≡3−4t (mod 6), holds modulo [m]Φ_m(q)^2.

Background

The paper proves a one-parameter q-supercongruence corresponding to Van Hamme-type (E.2) congruences modulo Φ_m(q)^3. The authors state that a stronger version should hold, based on numerical evaluation. Specifically, they conjecture that the congruence displayed as Theorem 1 remains valid modulo the product of the q-integer [m] and the square of the m-th cyclotomic polynomial Φ_m(q).

The authors further explain that specializing m=p^r and taking q→1 would yield a p-adic generalization of the paper’s hypergeometric supercongruence for all prime powers p^r satisfying p^r≡3−4t (mod 6), with t∈{1,2}.