  The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995)
is a two-parameter quadratic inequality for the number of linear extensions of
a poset $P= (X, \prec)$ with given value differences on three distinct elements
in $X$. We give two different proofs of this inequality for posets of width
two. The first proof is algebraic and generalizes CPC to a four-parameter
family. The second proof is combinatorial and extends CPC to a $q$-analogue.
Further applications include relationships between CPC and other poset
inequalities, including a new $q$-analogue of the Kahn--Saks inequality.
