  Recently, a strong exponential character bound has been established in [3]
for all elements $g \in \mathbf{G}^F$ of a finite reductive group
$\mathbf{G}^F$ which satisfy the condition that the centraliser
$C_{\mathbf{G}}(g)$ is contained in a $(\mathbf{G},F)$-split Levi subgroup
$\mathbf{M}$ of $\mathbf{G}$ and that $\mathbf{G}$ is defined over a field of
good characteristic. In this paper, assuming a weak version of Lusztig's
conjecture relating irreducible characters and characteristic functions of
character sheaves holds, we considerably generalize this result by removing the
condition that $\mathbf{M}$ is split. This assumption is known to hold whenever
$Z(\mathbf{G})$ is connected or when $\mathbf{G}$ is a special linear or
symplectic group and $\mathbf{G}$ is defined over a sufficiently large finite
field.
