Nonnegativity conjecture for the cd-index of regular CW spheres

A regular CW sphere is a regular CW complex whose underlying space is a sphere, and its $cd$-index is the polynomial $\Phi_P(c,d)$ encoding its flag $f$-vector, with $c$ of degree $1$ and $d$ of degree $2$. Nonnegativity conjecture. The coefficients in the $cd$-index of every regular CW sphere are nonnegative. This extends the nonnegativity known for convex polytopes to all regular CW spheres; the source gives no resolution of the conjecture.