An Invariant Subbundle of the KZ Connection mod $p$ and Reducibility of $\widehat{\mathfrak{sl}_2}$ Verma Modules mod $p$

We consider the KZ differential equations over $\mathbb C$ in the case, when its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb{F}_p$. We study the space of polynomial solutions of these differential equations over $\mathbb{F}_p$, constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $p$. We describe the algebraic equations for that subbundle and argue that the equations correspond to highest weight vectors of the associated $\widehat{\mathfrak{sl}_2}$ Verma modules over the field $\mathbb{F}_p$.