Finite generation of the Eulerian flag-vector cone

For each positive integer $n$, consider the closed cone of flag $f$-vectors of Eulerian posets of rank $n+1$.

Finite-generation conjecture. For every positive integer $n$, this closed cone is finitely generated.

The paper proves that doubled limit posets associated with even interval systems generate extreme rays, but notes that this is far from a complete description of the extreme rays. Finite generation would provide a finite extremal description of the Eulerian flag-vector cone in every rank.