The full-root and maximal-antichain bijection conjecture

Let $\Phi$ be a finite root system of rank $n$, with positive roots ordered by root order. A full root is a positive root whose support is the full Dynkin diagram, and an antichain has no two comparable elements. Full-root bijection conjecture. There should be a natural bijection between the full roots and the antichains of cardinality $n-1$ containing no simple roots. The paper obtains this as a reformulation of the preceding type-preserving duality conjecture, using that antichains of cardinality $n-1$ have full type exactly when they contain no simple roots. Its status is not resolved in the supplied text.