Bandari–Bayati–Herzog conjecture on homological shift ideals

Let $R=\mathbb{k}[x_1,\ldots,x_p]$ be a polynomial ring over a field $\mathbb{k}$, and let $\mathscr{P}$ be a polymatroid on $[p]$ with cage $\mathbf{m}\in\mathbb{N}^p$. Let $I_\mathscr{P}\subset R$ be its polymatroidal ideal, and for each $i\geq 0$ let ${\rm HS}_i(I_\mathscr{P})\subset R$ be the $i$-th homological shift ideal of $I_\mathscr{P}$. Bandari–Bayati–Herzog conjecture. All the homological shift ideals ${\rm HS}_i(I_\mathscr{P})$ of $I_\mathscr{P}$ are again polymatroidal ideals. The conjecture has been verified when $\mathscr{P}$ is a matroid, when it satisfies the strong exchange property, and when it has rank two; the paper settles it in general.