  For a smooth variety $Y$ over a perfect field of positive characteristic, the
sheaf $D_Y$ of crystalline differential operators on $Y$ (also called the sheaf
of $PD$-differential operators) is known to be an Azumaya algebra over
$T^*_{Y'},$ the cotangent space of the Frobenius twist $Y'$ of $Y.$ Thus to a
sheaf of modules $M$ over $D_Y$ one can assign a closed subvariety of
$T^*_{Y'},$ called the $p$-support, namely the support of $M$ seen as a sheaf
on $T^*_{Y'}.$ We study here the family of $p$-supports assigned to the
reductions modulo primes $p$ of a holonomic $\mathcal{D}$-module. We prove that
the Azumaya algebra of differential operators splits on the regular locus of
the $p$-support and that the $p$-support is a Lagrangian subvariety of the
cotangent space, for $p$ large enough. The latter was conjectured by
Kontsevich. Our approach also provides a new proof of the involutivity of the
singular support of a holonomic $\mathcal{D}$-module, by reduction modulo $p.$
