  In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of
all unipotent characters of finite groups of Lie type have been explicitly
determined except for six cases in groups of type $F_4$, $E_7$ and $E_8$. In
this paper, we show that the Schur indices of all cuspidal unipotent characters
for type $F_4$ and $E_8$ are~1, assuming that the group is defined over a field
of ``good'' characteristic. This settles four out of the six open cases. For
type $E_7$, we show that the Schur indices are at most~2.
