  Heath and Pemmaraju conjectured that the queue-number of a poset is bounded
by its width and if the poset is planar then also by its height. We show that
there are planar posets whose queue-number is larger than their height,
refuting the second conjecture. On the other hand, we show that any poset of
width $2$ has queue-number at most $2$, thus confirming the first conjecture in
the first non-trivial case. Moreover, we improve the previously best known
bounds and show that planar posets of width $w$ have queue-number at most
$3w-2$ while any planar poset with $0$ and $1$ has queue-number at most its
width.
