The type-preserving duality conjecture for full antichains

Let $\Phi$ be a finite root system of rank $n$, with positive roots $\Phi_{\geq 0}$ ordered by root order. An antichain is a set of pairwise incomparable positive roots. For an antichain $\mathsf{A}$, let $t(\mathsf{A})$ be the set of Dynkin-diagram edges whose endpoints occur together in the support of some root in $\mathsf{A}$. Call $\mathsf{A}$ of full type when $t(\mathsf{A})$ is the complete Dynkin diagram. Type-preserving duality conjecture. There should be a natural bijection between the antichains of cardinality $1$ and full type and the antichains of cardinality $n-1$ and full type. This is presented as a special case of Panyushev's proposed duality preserving type; the supplied text notes that the relevant duality had been constructed only for types $A$, $B$, and $C$.