The absolute-boundary conjecture for distributive-lattice graded graphs

Let $Y$ be a poset, let $L(Y)$ be the distributive lattice of all finite ideals of $Y$, and let $\Gamma$ be the Hasse diagram of $L(Y)$. Consider the space of all minimal infinite ideals of $Y$, and call a function on this space monotone positive if it is positive and monotone. Absolute-boundary conjecture. The absolute boundary of $\Gamma$ is the set of all monotone positive functions $f\leq 1$ on the space of all minimal infinite ideals of $Y$. This conjecture seeks a description of the absolute boundary for graded graphs arising as Hasse diagrams of distributive lattices, extending the preceding Pascal-graph examples. The supplied text gives no resolution or additional hypotheses.