Robustly transitive maps with critical points and large dimensional central spaces

Given any triplet of positive integers $n\ge 2$, $m$ and $k$ such that $n=m+k$, we exhibit a $C^1$ robustly transitive endomorphism of ${\mathbb{T}}^n$ with persistent critical points in the isotopy class of $F\times Id$, where $F$ is an expanding map of ${\mathbb{T}}^m$ and $Id$ is the identity of ${\mathbb{T}}^k$. Furthermore, if $k$ is small, the map is not only in the isotopy class but in fact a perturbation of $F\times Id$.