On $mathfrak{sl}_2$-triples for classical algebraic groups in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n in mathbb Z_{>0} $, and take $G$ to be one of the classical algebraic groups $mathrm{GL}_n(k)$, $mathrm{SL}_n(k)$, $mathrm{Sp}_n(k)$, $mathrm O_n(k)$ or $mathrm{SO}_n(k)$, with $mathfrak g = operatorname{Lie} G$. We determine the maximal $G$-stable closed subvariety $mathcal V$ of the nilpotent cone $mathcal N$ of $mathfrak g$ such that the $G$-orbits in $mathcal V$ are in bijection with the $G$-orbits of $mathfrak{sl}_2$-triples $(e,h,f)$ with $e,f in mathcal V$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on $mathfrak{sl}_2$-triples hold for classical algebraic groups over an algebraically closed field of ``small odd characteristic.