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The variation of the monodromy group in families of stratified bundles in positive characteristic

In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature conjecture stated and proved by Esnault and Langer in "On a positive equicharacteristic variant of the $p$-curvature conjecture" (Doc. Math. 18 (2013)). The main result is that if the ground field is uncountable then the strong form holds. In the case where the ground field is countable we provide positive and negative answers to possible generalizations.