  The queue-number of a poset is the queue-number of its cover graph viewed as
a directed acyclic graph, i.e., when the vertex order must be a linear
extension of the poset. Heath and Pemmaraju conjectured that every poset of
width $w$ has queue-number at most $w$. Recently, Alam et al. constructed
posets of width $w$ with queue-number $w+1$. Our contribution is a construction
of posets with width $w$ with queue-number $\Omega(w^2)$. This asymptotically
matches the known upper bound.
