The generalized cross-product inequality for posets

Assume the conditions of the generalized cross-product conjecture, so that $F(i,j)$ counts the relevant linear extensions for integer indices. Generalized cross-product inequality. For all indices satisfying $i\leq k$ and $j\leq\ell$,

$$F(i,j)F(k,\ell)\leq F(i,\ell)F(k,j).$$

This is presented as an even more general form of the preceding conjecture. The supplied text does not establish it in full generality, so its status is open.