  We investigate which higher rank simple Lie groups admit profinitely but not
abstractly commensurable lattices. We show that no such examples exist for the
complex forms of type $E_8$, $F_4$, and $G_2$. In contrast, there are
arbitrarily many such examples in all other higher rank Lie groups, except
possibly $\mathrm{SL}_{2n+1}(\mathbb{R})$, $\mathrm{SL}_{2n+1}(\mathbb{C})$,
$\mathrm{SL}_n(\mathbb{H})$, or groups of type $E_6$.
