The strong $\pi$-Sylow theorem for finite simple groups of Lie type of rank $1$

Let $\pi$ be a set of primes. A finite group is said to be a $\pi$-group if all prime divisors of its order belong to $\pi$. Following Wielandt, we say that for a finite group $G$ the $\pi$-Sylow theorem holds if all maximal $\pi$-subgroups of $G$ are conjugate; if the $\pi$-Sylow theorem holds for every subgroup of $G$, then $G$ is said to satisfy the strong $\pi$-Sylow theorem. The question of which finite nonabelian simple groups satisfy the strong $\pi$-Sylow theorem was posed by Wielandt in 1979. This paper completes an arithmetic description of the groups of Lie type of rank $1$ that satisfy the strong $\pi$-Sylow theorem.