Aluffi–Chen–Marcolli real-rootedness conjecture for the moduli space of stable pointed curves

Let $\overline{\mathcal{M}}_{0,n+1}$ be the moduli space of stable curves of genus zero with $n+1$ marked points, and let $P_{\overline{\mathcal{M}}_{0,n+1}}(t)$ denote its Poincaré polynomial. Equivalently, for the braid matroid $\mathcal{B}raid_n$ with minimal building set $\mathcal{G}_{\min}$, write

$$P_{\overline{\mathcal{M}}_{0,n+1}}(t)=\mathrm{H}(\mathcal{B}raid_n,\mathcal{G}_{\min})(t).$$

Aluffi–Chen–Marcolli conjecture. The polynomial $P_{\overline{\mathcal{M}}_{0,n+1}}(t)$ is real-rooted.

Real-rootedness is one of several positivity properties studied for Hilbert–Poincaré series of wonderful varieties associated with matroids and building sets. The source reports extensive computer evidence for this conjecture, while the claim is presented as open here.