  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.
