  Let $\mathsf{J(C)}$ be the poset of order ideals of a cominuscule poset
$\mathsf{C}$ where $\mathsf{C}$ comes from two of the three infinite families
of cominuscule posets or the exceptional cases. We show that the
Auslander-Reiten translation $\tau$ on the Grothendieck group of the bounded
derived category for the incidence algebra of the poset $\mathsf{J(C)}$, which
is called the \emph{Coxeter transformation} in this context, has finite order.
Specifically, we show that $\tau^{h+1}=\pm id$ where $h$ is the Coxeter number
for the relevant root system.
