Lattice conjecture for inverse atom sets of involutions in the symmetric group

Let $S_n$ be the symmetric group, let $\mathcal{I}(S_n)$ denote its set of involutions, and for $z\in\mathcal{I}(S_n)$ let $\mathcal{A}(z)$ be the corresponding set of atoms, equipped with the atomic order $<_A$. Lattice conjecture. If $z\in\mathcal{I}(S_n)$, then

$$(\mathcal{A}(z)^{-1},<_A)$$

is a lattice. This is presented as a related conjecture about the additional structure of atomic orders; determining the sets $\mathcal{A}(z)$ efficiently for involutions in arbitrary Coxeter groups remains an open problem.