Nuclear dimension for an inclusion of unital C*-algebras

Let $P \subset A$ be an inclusion of separable unital C*-algebras with finite Watatani index. Suppose that $E \colon A \rightarrow P$ has the Rokhlin property, that is, there is a projection $e \in A' \cap A^\infty$ such that $E^\infty(e) = ({\rm Index}E)^{-1}1$. We show that if $A$ has nuclear dimension $n$, then $P$ has nuclear dimension less than or equal to $n$. In particular, if an action $\alpha$ of a finite group $G$ on $A$ has the Rokhlin property, then the nuclear dimension of the crossed product algebra $A \rtimes_\alpha G$ is less than or equal to that of $A$.