Cook–Nagel and Constantinescu–Varbaro conjecture on flag-complex f-vectors

Let an $f$-vector of a simplicial complex record its face numbers, and let an $h$-vector be the usual transform of its $f$-vector. A simplicial complex is flag if every minimal non-face has cardinality two, balanced if its vertices admit a proper coloring with one color for each dimension, and vertex decomposable in the usual recursive sense.

Cook–Nagel and Constantinescu–Varbaro conjecture. The following equality of sets holds:

$$\left\{ \begin{array}{c} \text{$f$-vectors of } \\ \text{flag complexes} \end{array} \right\} = \left\{ \begin{array}{c} \text{$h$-vectors of balanced}\\ \text{vertex decomposable flag complexes} \end{array} \right\}.$$

This asks whether the equivalence between $f$-vectors of simplicial complexes and $h$-vectors of balanced vertex decomposable complexes remains valid after restricting to flag complexes. The source attributes the conjecture to Cook and Nagel, and to Constantinescu and Varbaro; no resolution is given in the supplied text.