Toward a sharp Baer–Suzuki theorem for the $\pi$-radical: exceptional groups of small rank

Let $\pi$ be a proper subset of the set of all prime numbers. Denote by $r$ the least prime number not in $\pi$, and put $m=r$, if $r=2,3$, and $m=r-1$ if $r\geqslant 5$. We look at the conjecture that a conjugacy class $D$ in a finite group $G$ generates a $\pi$-subgroup in $G$ (or, equivalently, is contained in the $\pi$-radical) iff any $m$ elements from $D$ generate a $\pi$-group. Previously, this conjecture was confirmed for finite groups whose every non-Abelian composition factor is isomorphic to a sporadic, alternating, linear or unitary simple group. Now it is confirmed for groups the list of composition factors of which is added up by exceptional groups of Lie type ${}^2B_2(q)$, ${}^2G_2(q)$, $G_2(q)$, and ${}^3D_4(q)$.