Seymour's half-order matching conjecture

Let $G$ be a finite simple graph, and let a model consist of pairwise vertex-disjoint connected branch sets, with adjacent branch sets joined by an edge. Seymour's half-order matching conjecture. Every graph $G$ with $\alpha(G)=2$ has a

$$K_{\lceil |V(G)|/2\rceil}$$

-model in which each branch set has size at most $2$. The paper presents this as an implication of Seymour's matching strengthening and later establishes equivalence, while the conjecture remains open.