Keller maps and holonomic $D$-modules

Let $\text{kk}$ be a field of characteristic $p$ and $R=\text{kk}[x_1,\dots ,x_n]$. Let $D$ denote the ring of $\text{kk}$-linear differential operators on $R$. Let $\phi :R\to R$ be a $\text{kk}$-algebra endomorphism whose Jacobian is a unit in $R$. We show that there is a unique $\text{kk}$-algebra endomorphism $\mathrm{\Phi }:D\to D$ such that $\mathrm{\Phi }{\mathrm{\mid }}_R=\phi$ and that the restriction of scalar via $\mathrm{\Phi }$ preserves holonomicity. We also construct an example of a $\text{kk}$-algebra endomorphism of $D$ that does not preserve holonomicity.