Flat endomorphisms for mod $p$ equivariant quantum connections from quantum Steenrod operations

We present a method for constructing covariantly constant endomorphisms for the mod $p$ equivariant quantum connection, using the quantum Steenrod power operations of Fukaya and Wilkins. The example of the cotangent bundle of the projective line is fully computed, and we discuss the relationship with the mod $p$ solutions of trigonometric KZ equation recently constructed by Varchenko. As a byproduct, we compute the first examples of quantum Steenrod operations that are not a priori determined by ordinary Gromov--Witten theory and classical Steenrod operations, which may be of independent interest.