Aluffi's covolume conjecture for adjoint polynomials of polyhedral cones

Let $C\subseteq\mathbb{R}^{d+1}$ be a polyhedral cone whose cross section with the hyperplane $\{x_0=1\}$ is a polytope $P\subseteq\mathbb{R}^d$. The adjoint polynomial of $C$ is the polynomial obtained from a triangulation $T$ of $C$ by

$$adj_C(t)=\sum_{\sigma\in T}vol(\sigma)\prod_{v\in V(C)\backslash V(\sigma)}(v\cdot t),$$

where $V(C)$ and $V(\sigma)$ are the sets of vertex rays and $vol(\sigma)$ is normalized volume. Aluffi's covolume conjecture. If the polytope $P$ lies in the positive orthant, then the adjoint polynomial of $C$ is a covolume polynomial. The paper proves this by identifying the adjoint polynomial with a multidegree polynomial and applying the result that all multidegree polynomials with arbitrary $\mathbb{N}^{d+1}$-gradings are covolume polynomials; thus the conjecture is resolved.