Bounding regularity of $F{I}^m$-modules

Let $FI$ be a skeleton of the category of finite sets and injective maps, and $F{I}^m$ the product of $m$ copies of $FI$. We prove that if an $F{I}^m$-module is generated in degree $⩽d$ and related in degree $⩽r$, then its regularity is bounded above by a function of $m$, $d$, and $r$.