The order-scattered ideal-lattice conjecture for join-semilattices

Let $P$ be a join-semilattice, and let $\operatorname{Id} P$ denote its lattice of ideals. An ordered set is order-scattered if it does not contain a copy of the order of the rationals. Write $\powerset^{<\omega}(\mathbb N)$ for the finite subsets of the natural numbers, ordered by inclusion, and let $\Omega(\eta)$ be the ordered structure used in the stated embeddability condition. Order-scattered ideal-lattice conjecture. $\operatorname{Id} P$ is order-scattered if and only if $P$ is order-scattered, and neither $\powerset^{<\omega}(\mathbb N)$ nor $\Omega(\eta)$ is embeddable into $P$ as a join-semilattice. This conjecture was stated in the cited work and has been proved, so the characterization is now a theorem.