Finding all solutions to the KZ equations in characteristic p

The KZ equations are differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional conformal field theories associated with an affine Lie algebra at a fixed level. They form a system of complex partial differential equations with regular singular points satisfied by the n-point functions of affine primary fields. In [SV1] the KZ equations were identified with equations for flat sections of suitable Gauss-Manin connections, and solutions of the KZ equations were constructed in the form of multidimensional hypergeometric integrals. In [SV2] the KZ equations were considered modulo a prime number p, and polynomial solutions of the KZ equations modulo p were constructed by an elementary procedure as suitable p-approximations of the hypergeometric integrals. In this paper we address the problem of whether all solutions of the KZ equations modulo p are generated by the p-hypergeometric solutions. We consider the first nontrivial example of the KZ equations and demonstrate that, indeed, in this case, all solutions of the KZ equations modulo p stem from the p-hypergeometric solutions.