Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds

Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X{)}^4\le 648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X{)}^4\le 7332$. Under the additional assumption that a primitive Weil polarization is Cartier outside finitely many points, the latter bound improves to the sharp bound $1024$; if its non-Cartier locus has dimension at most one, we obtain $6084$.