Castillo–Cid-Ruiz–Mohammadi–Montaño conjecture on Möbius supports

Let $\mathscr{P}$ be a polymatroid on $[p]$, let $I(\mathscr{P})$ be its independence polytope, and let $\mu_\mathscr{P}:\mathbb{Z}^p\to\mathbb{Z}$ be its Möbius function. Define the Möbius support by

$$\mu\text{-supp}(\mathscr{P})=\{\mathbf{n}\in\mathbb{N}^p\mid \mu_\mathscr{P}(\mathbf{n})\neq 0\}.$$

Castillo–Cid-Ruiz–Mohammadi–Montaño conjecture. The Möbius support of $\mathscr{P}$ is a generalized polymatroid, that is, a homogenization of it yields a polymatroid. The paper settles this conjecture, so the asserted property is known in general.