A new conjecture about filters

Let \mathfrak{F}(S) denotes the set of filters on a poset S, ordered reversely to set theoretic inclusion of filters. Let Da for a lattice element a denote its sublattice \{ x \mid x \leq a \}. Let Z(X) denotes the set of complemented elements of the lattice X.

Conjecture \mathfrak{F}(Z(D\mathcal{A})) is order-isomorphic to D\mathcal{A} for every filter \mathcal{A} on a set. If they are isomorphic, find an isomorphism.

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