On the exponent of geometric unipotent radicals of pseudo-reductive groups

Let $k/k$ be a finite purely inseparable field extension and let $G$ be a reductive $k$-group. We denote by $G=R_{k/k}(G)$ the Weil restriction of $G$ across $k/k$, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical $mathscr{R}_{u}(G_{bar{k}})$, focusing on the case $G=GL_n$.