A new conjecture about filters

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

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