The cd-index flag f-vector conjecture for regular CW-spheres

Let $P$ be an $(n-1)$-dimensional regular CW-sphere, or more generally a Gorenstein* poset of rank $n+1$. Write its homogeneous cd-index as

$$\Phi_P=\sum_w a_w w,$$

where $w$ ranges over cd-monomials of degree $n$. For a cd-monomial $w$, let $F_w\subseteq[n-1]$ be the associated subset with no two consecutive elements, and define $\alpha_S(\Phi_P)=a_w$ when $S=F_w$, and $\alpha_S(\Phi_P)=0$ otherwise. A colored simplicial complex has flag $f$-vector $(f_S(\Delta):S\subseteq[n-1])$, where $f_S(\Delta)$ counts faces whose vertex colors are exactly $S$. The cd-index flag f-vector conjecture. There exists an $(n-1)$-colored simplicial complex $\Delta$ such that

$$f_S(\Delta)=\alpha_S(\Phi_P)$$

for every $S\subseteq[n-1]$. Thus the cd-index should itself be the flag $f$-vector of a colored complex. This conjecture would extend the authors' results from S*-shellable spheres to all regular CW-spheres and Gorenstein* posets; its general status is open.