In the setting of Theorem B, let $Z=D$ be an irreducible component of $\mathscr{X}_{k}$. Then there is a natural $\mathbb{Z}$-linear embedding of $\Star(v_{D})$ inside the fan $\Sigma_{D}$ of $D$ which sends the polyhedral decomposition of $\Star(v_{D})$ to the cone decomposition of $\Sigma_{D}$.