The Eulerian conjecture for the augmented wiring-diagram poset

Let $W_n$ be the poset of equivalence classes of lensless wiring diagrams, ordered by breaking crossings, and let $\widehat{0}$ be a new minimal element adjoined to $W_n$. Eulerian conjecture. The poset $W_n\cup\{\widehat{0}\}$ is Eulerian. The source states that this conjecture implies the Eulerian conjecture for $EP_n$. Its resolution is not given in the supplied text.