Finite-width characterization of locally-finite distributive lattices with few ideal components

Let $\mathcal{L}$ be a locally-finite distributive lattice, and let $\mathcal{P}$ be its set of prime filters. Let $\mathcal{I}$ be the lattice of ideals of $\mathcal{P}$. A lattice has finite width when it contains no infinite antichains. Finite-width characterization. The lattice $\mathcal{I}$ contains at most three connected components: one isomorphic to $\mathcal{L}$, perhaps one containing the empty ideal, and perhaps one containing the ideal of all prime filters, if and only if $\mathcal{L}$ has finite width. This characterization is motivated by examples relevant to constructing RSK algorithms, including $\mathbb{Z}\times\mathbb{Z}$, $\mathbb{B}_{\mathrm{fin}}$, and skew-strip lattices. Determining whether the stated equivalence holds for all locally-finite distributive lattices remains open.