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We show that Shakirov's non-stationary difference equation, when it istruncated, implies the quantum Knizhnik-Zamolodchikov ($q$-KZ) equation for$U_{\mathsf v}(A_1^{(1)})$ with generic spins. Namely we can tune massparameters so that the Hamiltonian acts on the space of finite Laurentpolynomials. Then the representation matrix of the Hamiltonian agrees with the$R$-matrix, or the quantum $6j$ symbols. On the other hand, we prove that the$K$ theoretic Nekrasov partition function from the affine Laumon space isidentified with the well-studied Jackson integral solution to the $q$-KZequation. Combining these results, we establish that the affine Laumonpartition function gives a solution to Shakirov's equation, which was aconjecture in our previous paper. We also work out the base-fiber duality andfour dimensional limit in relation with the $q$-KZ equation.

Non-stationary difference equation and affine Laumon space II: Quantum Knizhnik-Zamolodchikov equation