gamma-vectors of edge subdivisions of the boundary of the cross polytope

For any flag simplicial complex $\mathrm{\Theta }$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $\mathrm{\Gamma }(\mathrm{\Theta })$ (dependent on the sequence of subdivisions) whose $f$-vector is the $\gamma$-vector of $\mathrm{\Theta }$. This proves that the $\gamma$-vector of any such simplicial complex satisfies the Frankl-F"{u}redi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We show that when $\mathrm{\Theta }$ is the dual simplicial complex to a nestohedron, and the sequence of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that $\mathrm{\Gamma }(\mathrm{\Theta })$ is equal to the flag simplical complex defined there.