Equality of the Eulerian and doubled half-Eulerian flag-vector cones

Let ${\cal C}^{n+1}_{\cal E}$ be the closed cone of flag vectors of Eulerian posets of rank $n+1$, and let ${\cal C}^{n+1}_{\cal D}$ be the closed cone of flag vectors of horizontal doubles of half-Eulerian posets of rank $n+1$.

Cone equality conjecture. The two cones are equal:

$${\cal C}^{n+1}_{\cal E}={\cal C}^{n+1}_{\cal D}.$$

For low ranks the cones coincide, and no Eulerian poset is known whose flag vector lies outside the cone generated by doubled half-Eulerian posets. The conjecture would identify all closed-cone limits of Eulerian flag vectors with those arising from horizontal doubles.