A strong version of the Sims conjecture for primitive parabolic permutation representations of finite simple groups Lie types $G_2, F_4$ and $E_6$

For a finite group $G$, subgroups $M_1$ and $M_2$ of $G$ and any $i\in\mathbb{N}$, the subgroups $(M_1, M_2)^i$ and $(M_2, M_1)^i$ of $M_1\cap M_2$ are defined, inductively on $i$, as follows:
$$(M_1, M_2)^1 = (M_1\cap M_2)_{M_1},~(M_2, M_1)^1 = (M_1\cap M_2)_{M_2},$$

$$(M_1, M_2)^{i+1} = ((M_2, M_1)^i)_{M_1},~(M_2,M_1)^{i+1} = (M_1,M_2)^i_{M_2}.$$
Here, for $H\leq G$, $H_G$ denotes $\bigcap_{g\in G}gHg^{-1}$. Denote by $\Pi$ the set of all triples $(G,M_1,M_2)$ such that $G$ is a finite group, $M_1$ and $M_2$ are distinct conjugate maximal subgroups of $G$, $(M_1)_G=(M_2)_G=1$, and $1 < |(M_1,M_2)^{2}| \leq |(M_2,M_1)^{2}|$. The triples $(G,M_1,M_2)$ and $(G',M'_1,M'_2)$ from $\Pi$ are equivalent if there exists an isomorphism from $G$ to $G'$ mapping $M_1$ to $M'_1$ and $M_2$ to $M'_2$. The present paper is a continuation of the investigations by A.S. Kondrat'ev and V.I. Trofimov on a description of the set $\Pi$. It is obtained the description up to equivalence all triples $(G,M_1,M_2)$ from $\Pi$ in the case when $G$ is a finite simple group of Lie type $G_2$, $F_4$ or $E_6$, and $M_1$ is a parabolic maximal subgroup of $G$.