Nonhomogeneous analytic families of trees

We consider a dichotomy for analytic families of trees stating that either there is a colouring of the nodes for which all but finitely many levels of every tree are nonhomogeneous, or else the family contains an uncountable antichain. This dichotomy implies that every nontrivial Souslin poset satisfying the countable chain condition adds a splitting real. We then reduce the dichotomy to a conjecture of Sperner Theory. This conjecture is concerning the asymptotic behaviour of the product of the sizes of the m-shades of pairs of cross-t-intersecting families.