О сопряженности $\mathrm{Alt}_5$-подгрупп из подгруппы Боровика группы $E_8(q)$

Let $p \geq 7$ be a prime, $q = p^n$, where $n \in {\mathbb N}$, and $k$ be the algebraic closure of the field $\mathbb{F}_q$. Let $G \cong E_8(k)$ be a simple linear algebraic group of type $E_8$ over the field $k$, and $\sigma : G \rightarrow G$ be a Steinberg endomorphism of $G$ such that $G_{\sigma} \cong E_8(q)$. Let $M \cong (\mathrm{Alt}_5 \times \mathrm{Sym}_6).2$ be a Borovik subgroup of the group $G$ and $M < G_{\sigma}$. An open question is whether the normal $\mathrm{Alt}_5$-subgroup of $M$ and a diagonal $\mathrm{Alt}_5$-subgroup of $\mathrm{soc}(M)$ are conjugated in $G_\sigma$ or not.
In 1998, D. Frey investigated conjugated classes of $\mathrm{Alt}_5$-subgroups in $E_8(\mathbb{C})$. But, description of the classes with zero-dimensional centralizers was not obtained. In particular, it was not clear are $\mathrm{Alt}_5$-subgroups of a Borovik subgroup of $E_8(\mathbb{C})$ with zero-dimensional centralizers conjugated in $E_8(\mathbb{C})$ or not. This problem was solved by G. Lusztig in 2003. Actually, the Lusztig result is more general and concerns regular homorphisms from $\mathrm{Alt}_5$ to connected reductive algebraic group over an algebraically closed field $k'$ of characteristic $p$ where $p=0$ or $p \geq 7$. The Lusztig result implies, in particular, that $\mathrm{Alt}_5$-subgroups of a Borovik subgroup of $E_8(k')$ with zero-dimensional centralizers are conjugated in $E_8(k')$. We use the Lusztig result to prove that the normal $\mathrm{Alt}_5$-subgroup of the group $M$ is conjugated with a diagonal $\mathrm{Alt}_5$-subgroup of $\mathrm{soc}\,(M)$ in $G_{\sigma^m}$ where $m \leq 6$.