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 $G'=GL_n$.