The h-vector characterization conjecture for flag Cohen–Macaulay complexes

Let $h(\Delta)$ denote the $h$-vector of a simplicial complex and $f(\Delta)$ its $f$-vector. Flag Cohen–Macaulay h-vector conjecture. The set of $h$-vectors of flag Cohen–Macaulay simplicial complexes equals the set of $f$-vectors of flag simplicial complexes. The source says this conjecture implies the preceding flag Cohen–Macaulay conjecture and gives no resolution.