A family of indecomposable rank-$n$ vector bundles on ${\mathbb{P}}^n\times {\mathbb{P}}^n$ in positive characteristics

We construct a sequence of rank-$n$ indecomposable vector bundles on ${\mathbb{P}}^n\times {\mathbb{P}}^n$ for every $n\ge 2$ and in every positive characteristic that are not pullbacks via any map ${\mathbb{P}}^n\times {\mathbb{P}}^n\to {\mathbb{P}}^m$