  Humphreys' conjecture on blocks parametrises the blocks of reduced enveloping
algebras $U_\chi({\mathfrak g})$, where ${\mathfrak g}$ is the Lie algebra of a
reductive algebraic group over an algebraically closed field of characteristic
$p>0$ and $\chi\in{\mathfrak g}^{*}$. It is well-known to hold under Jantzen's
standard assumptions. We note here that it holds under slightly weaker
assumptions, by utilising the full generality of certain results in the
literature. We also provide a new approach to prove the result for ${\mathfrak
g}$ of type $G_2$ in characteristic 3, a case in which the previously mentioned
weaker assumptions do not hold. This approach requires some dimensional
calculations for certain centralisers, which we conduct in the Appendix for all
the exceptional Lie algebras in bad characteristic.
